B   14   555 


THE  INTERNATIONAL  SCIENTIFIC  SERIES. 
VOLUME  XXXVIII. 


THE  INTERNATIONAL  SCIENTIFIC   SERIES. 


THE 


CONCEPTS  AND   THEORIES 


OF 


MODERN  PHYSICS. 


J.  B.  STALLO. 


NEW  YORK: 
D.    APPLETON    AND    COMPANY, 

1,    3,    AND    5    BOND     STKEET. 

1882. 


COPYRIGHT  BY 

D.  APPLETON  AND  COMPANY, 


7O 

JS7 


OF 

DR.    WALTER    STALLO 


summa  tamen  omnia  constant. 


"  Inficitur  autem  intellectus  humanus  ex  intuitu  eorum,  quae 
in  artibus  mechanicis  fiunt,  in  quibus  corpora  per  compositiones  aut 
separatio7ies  utplurimum  alterantur ;  ut  cogitet,  simile  quiddam 
etiam  in  naturd  rerum  universali  fieri.  Undefluxit  commentum 
illud  elementorum  deque  illorum  concursu,  ad  constituenda  corpora 
naturalia" 

BACO  VEEUL.,  Nov.  OEG.,  LIB.  I,  APH.  66. 


PREFACE 


THE  following  pages  are  designed  as  a  contribution, 
not  to  physics,  nor,  certainly,  to  metaphysics,  but  to  the 
theory  of  cognition.  Their  contents  are  the  result  of 
a  somewhat  careful  study  of  the  true  relation  of  the 
physical  sciences  to  the  general  progress  of  human 
knowledge.  It  is  the  common  opinion  of  contempo- 
rary physicists  that  there  was  a  total  breach  of  continu- - 
ity  in  the  line  of  this  progress  at  the  point  where  the 
thoughts  of  men  were  turned  from  ancient  and  mediae- 
val traditions  respecting  the  phenomena  of  nature  and 
their  significance  to  the  order  and  sequence  of  these 
phenomena  as  disclosed  by  their  own  observations  and 
experiments,  and  that  the  structure  of  what  may,  for 
want  of  a  better  name,  still  be  called  philosophy  now 
rests  upon  foundations  wholly  different  from  those  upon 
which  it  stood  before  the  days  of  Galilei  and  Bacon. 
According  to  this  view,  Bacon's  demand  (in  the  pref- 
ace to  his  Novum  Organum)  "  that  the  whole  work  of 
the  mind  be  undertaken  anew  " — ut  opus  mentis  uni- 
versum  de  integro  resumatur — has  been  thoroughly 
complied  with,  and  Newton's  admonition  to  the  physi- 
cists, "  to  beware  of  metaphysics,"  has  been  effectually  - 
heeded.  The  belief  is  that  modern  physical  science 
has  not  only  made  its  escape  from  the  cloudy  regions 
of  metaphysical  speculation,  and  discarded  its  methods 


8  PREFACE. 

of  reasoning,  but  that  it  has  likewise  emancipated  itself 
from  the  control  of  its  fundamental  assumptions.  It  is 
my  conviction  that  this  belief  is  but  partially  conform- 
able to  the  fact,  and  that  the  prevailing  misconceptions 
in  regard  to  the  true  logical  and  psychological  premisses 
of  science  are  prolific  of  errors,  whose  reaction  upon  the 
character  and  tendencies  of  modern  thought  becomes 
more  apparent  from  day  to  day.  The  shallow  and  scio- 
listic  materialism — I  allude,  of  course,  not  to  its  sup- 
posed ethical  but  to  its  purely  intellectual  aspects — 
which  for  a  time  threatened  to  blight  the  soil  and  poi- 
son the  atmosphere  even  of  the  old  highlands  of  thought 
on  the  continent  of  Europe,  claims  to  be  a  presentation 
of  conclusions  from  the  facts  and  principles  established 
in  the  several  departments  of  physical  science.  It  is 
part  of  my  endeavor  to  meet  this  claim  by  an  examina- 
tion of  the  fundamental  concepts  and  general  theories 
of  that  department  of  physical  science  which  is,  in  a 
sense,  the  basis  and  support  of  all  its  other  departments 
—the  department  of  physics.  It  will  be  seen  at  once, 
upon  a  most  cursory  glance  at  any  one  of  the  chapters 
of  this  little  book,  that  it  is  in  no  wise  intended  as  an 
open  or  covert  advocacy  of  a  return  to  metaphysical 
methods  and  aims ;  but  that,  on  the  contrary,  its  ten- 
dency is  throughout  to  eliminate  from  science  its  latent 
metaphysical  elements,  to  foster  and  not  to  repress  the 
spirit  of  experimental  investigation,  and  to  accredit  in- 
stead of  discrediting  the  great  endeavor  of  scientific 
research  to  gain  a  sure  foothold  on  solid  empirical 
ground,  where  the  real  data  of  experience  may  be  re- 
duced without  ontological  prepossessions.  An  attentive 
perusal  of  these  pages  will  make  it  clear,  I  think,  that 
this  endeavor  is  continually  thwarted  by  the  insidious 
intrusion  into  the  meditations  of  the  man  of  science 


PREFACE.  9 

of  the  old  metaphysical  spirit.  This  fact  having  been 
established,  it  was  incumbent  on  me  to  ascertain,  if 
possible,  its  causes  and,  within  the  narrow  limits  at  my 
command,  to  develop  its  consequences.  In  the  per- 
formance of  this  task  it  became  necessary — inasmuch 
as  I  wrote  for  a  class  of  readers  with  whom,  unfortu- 
nately, familiarity  with  the  laws  of  thought  is  a  some- 
wThat  rare  accomplishment — to  make  an  excursion  into 
the  domain  of  logic,  and  to  enter  upon  a  brief  discus- 
sion of  the  theory  of  conception.  This  discussion  is,  of 
necessity,  very  perfunctory,  but  I  venture  to  hope  that 
it  will  not  prove  wholly  devoid  of  interest  even  to  those 
who  are  thoroughly  familiar  with  the  subject.  Fur- 
thermore, the  atomo-mechanical  theory,  which  is  sup- 
posed to  be  the  only  and  all-sufficient  basis  of  the  sci- 
ence of  physics,  has  become  complicated  with,  or,  rather, 
has  led  to,  certain  remarkable  speculations  as  to  the 
nature  and  properties  of  space  ;  and  this  necessitated 
another  excursion  into  the  field  of  mathematics,  for  the 
purpose  of  examining  the  validity  of  the  doctrines  of 
what  is  generally  known  as  transcendental  geometry 
with  its  hypotheses  of  non-homaloidal  space  and  of 
space  of  more  than  three  dimensions. 

What  is  here  presented  is  not,  of  course,  a  new  the- 
ory of  the  universe,  or  a  novel  system  of  philosophy.  I 
have  undertaken,  not  to  solve  all  or  any  of  the  problems 
of  cognition,  but  simply  to  show  that  some  of  them  are 
in  need  of  being  stated  anew  so  as  to  be  rationalized,  if 
not  deepened.  It  is  an  old  truth,  which,  however,  is 
too  often  lost  sight  of,  that  many  of  the  questions  of 
science  and  philosophy  remain  unanswered,  not  by  rea- 
son of  the  insufficiency  of  our  knowledge,  but  because 
the  questions  themselves  are  founded  on  erroneous  as-,  / 
sumptions  and  require  answers  in  irrational  or  impos- 


10  PREFACE. 

sible  terms.  The  utter  anarchy  which  notoriously  pre- 
vails in  the  discussion  of  ultimate  scientific  questions, 
so  called,  indicates  that  a  determination  of  the  proper 
attitude  of  scientific  inquiry  toward  its  objects  is  the 
most  pressing  intellectual  need  of  our  time,  as  it  is  an 
indispensable  prerequisite  of  real  intellectual  progress 
at  all  times.  And  such  a  determination,  however  par- 
tial, is  in  itself  a  decided  advance  in  the  direction  of 
our  legitimate  cognitive  aspirations.  "  Kightly  to  pro- 
pose a  problem,"  says  "Whewell,  "  is  no  inconsiderable 
I  step  to  its  solution."  In  the  language  of  Kant :  "  Es  ist 
schon  ein  grosser  und  noethiger  Beweis  der  KlugJielt 
und  Einsicht  zu  wissen,  was  man  vernuenftiger  Weise 
fragen  solle."  And  in  the  pithy  phrase  of  Bacon: 
" Prudens  quaestio  quasi  dimidium  scientiae" 

My  views  respecting  the  actual  state  of  physical 
science  and  the  value  of  many  of  the  current  theoretical 
interpretations  of  scientific  facts  are,  no  doubt,  at  vari- 
ance with  the  tenets  of  many  distinguished  scientific 
men.  That  I  have,  nevertheless,  given  fearless  expres- 
sion to  them  will  not,  I  hope,  be  construed  as  a  want  of 
appreciation  of  the  merits  of  those  to  whose  labors  mod- 
ern culture  owes  its  life,  and  the  pursuit  of  knowledge  in 
the  interest  of  that  culture  its  practical  success.  And, 
if  it  should  be  regarded  as  evidence  of  presumption,  I 
desire  to  say  that  there  are  suggestions,  in  many  of  the 
utterances  of  the  men  of  science  here  referred  to,  of  a 
growing  sense  of  the  questionability  of  some  of  the  ele- 
ments of  their  scientific  faith.  I  have  taken  frequent 
occasion,  in  the  progress  of  my  discussion,  to  point  to 
these  suggestions,  to  the  end  of  showing  that  my 
thoughts  are,  after  all,  but  the  inevitable  outcome  of 
the  tendencies  of  modern  science,  and  are,  therefore, 
rather  " partus  teinporis  quam  ingenii" 


PREFACE.  11 

I  deem  it  important  to  have  it  understood,  at  the 
outset,  that  this  treatise  is  in  no  sense  a  further  exposi- 
tion of  the  doctrines  of  a  book  ("  The  Philosophy  of 
Nature,"  Boston,  Crosby  &  Nichols,  1848)  which  I 
published  more  than  a  third  of  a  century  ago.  That 
book  was  written  while  I  was  under  the  spell  of  Hegel's 
ontological  reveries — at  a  time  when  I  was  barely  of 
age  and  still  seriously  affected  with  the  metaphysical 
malady  which  seems  to  be  one  of  the  unavoidable  dis- 
orders of  intellectual  infancy.  The  labor  expended  in 
writing  it  was  not,. perhaps,  wholly  wasted,  and  there 
are  things  in  it  of  which  I  am  not  ashamed,  even  at 
this  day ;  but  I  sincerely  regret  its  publication,  which  is 
in  some  degree  atoned  for,  I  hope,  by  the  contents  of 
the  present  volume. 

It  ought  to  be  added  that  parts  of  the  seventh  and 
eleventh  chapters  of  this  book,  and  a  few  sentences  in 
the  other  chapters,  were  published  in  "  The  Popular 
Science  Monthly  "  in  October,  November,  and  Decem- 
ber, 1873,  and  January,  1874. 

J.  B.  STALLO. 

CINCINNATI,  September  1,  1881. 


CONTENTS. 


CHAPTER   I.  PAGB 

Introductory  ..." 15 

CHAPTER   II. 
First  Principles  of  the  Mechanical  Theory  of  the  Universe      .        .       25 

CHAPTER  III. 
The  Proposition  that  the  Elementary  Units  of  Mass  are  equal         .       30 

CHAPTER  IV. 

The  Proposition  that  the  Elementary  Units  of  Mass  are  absolutely 

hard  and  inelastic 40 

CHAPTER  V. 

The  Proposition  that  the  Elementary  Units  of  Mass  are  absolutely 

inert 52 

CHAPTER  VI. 

The  Proposition  that  all  Potential  Energy  is  in  Reality  kinetic. — 

Evolution  of  the  Doctrine  of  the  Conservation  of  Energy  .        .       66 

CHAPTER  VII. 
The  Theory  of  the  Atomic  Constitution  of  Matter   ....       84 

CHAPTER  VIII. 

The  Kinetic  Theory  of  Gases. — Conditions  of  the  Validity  of  Scien- 
tific Hypotheses        .        .        . 104 


14:  CONTENTS. 

CHAPTER  IX. 

PAGE 

The  Relation  of  Thoughts  to  Things. — The  Formation  of  Concepts. 

— Metaphysical  Theories  .         .         .         . .       .         .        .        .129 

CHAPTER  X. 

Character  and  Origin  of  the  Mechanical  Theory. — Its  Exemplifica- 
tion of  the  First  and  Second  Radical  Errors  of  Metaphysics  .  148 

CHAPTER   XI. 

Character  and  Origin  of  the  Mechanical  Theory  (continued). — Its 

Exemplification  of  the  Third  Radical  Error  of  Metaphysics        .     171 

CHAPTER   XII. 

Character  and  Origin  of  the  Mechanical  Theory  (continued). — Its 

Exemplification  of  the  Fourth  Radical  Error  of  Metaphysics      .     183 

CHAPTER  XIII. 

The  Theory  of  the  Absolute  Finitude  of  the  World  and  of  Space. — 
The  Assumption  of  an  Absolute  Maximum  of  Material  Existence 
as  a  Necessary  Complement  to  the  Assumption  of  the  Atom  as 
its  Absolute  Minimum. — Ontology  in  Mathematics. — The  Reifi- 
cation  of  Space.  —  Modern  Transcendental  Geometry.  —  Non- 
homaloidal  (Spherical  and  Pseudo-spherical)  Space  .  .  .  207 

CHAPTER   XIV. 

Metagcometrical   Space   in  the  Light  of   Modern  Analysis. — Rie- 

mann's  Essay 248 

CHAPTER   XV. 

Cosmological  and  Cosmogenetic  Speculations. — The  Nebular  Hy- 
pothesis   270 

CHAPTER  XVI. 

Conclusion  ,     294 


THE 

UNIVERSITY 


THE  CONCEPTS  AND  THEOEIES 


MODERN     PHYSICS. 


CHAPTEE  I. 

INTRODUCTORY. 

MODERN  physical  science  aims  at  a  mechanical  inter- 
pretation of  all  the  phenomena  of  the  universe.  It 
seeks  to  explain  these  phenomena  by  reducing  them  to 
the  elements  of  mass  and  motion  and  exhibiting  their 
diversities  and  changes  as  mere  differences  and  varia- 
tions in  the  distribution  and  aggregation  of  ultimate 
and  invariable  bodies  or  particles  in  space.  Naturally 
the  supremacy  of  mechanics  became  conspicuous  first 
in  the  domains  of  those  sciences  which  deal  with  the 
visible  motions  of  palpable  masses — in  astronomy  and 
molar  physics ;  but  its  recognition  is  now  all  but  uni- 
versal in  all  the  physical  sciences,  including,  not  only 
molecular  physics  and  chemistry,  but  also  such  depart- 
ments of  scientific  inquiry  as  are  conversant  about  the 
phenomena  o'f  organic  life. 

It  is  said  that  the  theoretical  no  less  than  the  prac- 
tical progress  of  the  natural  sciences,  -during  the  last 


16  CONCEPTS   OF   MODERN  PHYSICS.       - 

three  centuries,  is  an  achievement  of  mechanics  which, 
besides  devising  the  instruments  of  successful  scientific 
research,  has  also  supplied  its  principles  and  methods. 
It  is,  indeed,  incontestable  that  the  attempt  at  a  con- 
sistent application  of  mechanical  principles  marks  a  new 
epoch  in  the  history  of  science.  The  founders  of  mod- 
ern physics  proceeded  upon  the  tacit  if  not  upon  the 
declared  assumption  that  all  true  explanations  of  natural 
phenomena  are  mechanical  explanations.  That  this  did 
not  at  once  find  articulate  expression  is  due,  partly  to 
the  ,f act  that,  principles  are  wont  to  assert  themselves, 
in;  thought  'ds  ^'action,  before  they  are  distinctly  appre- 
cbe^3e*do  ate&jprartjy.'lo  £l>e  circumstance  that  science,  for 
a  long  time,  vvas  const  rained  to  flourish  under  the  shad- 
ows of  metaphysics  and  theology.""  But  it  was  not  long 
after  the  days  of  Stevinus,  FerhTat  and  Galilei  before 
the  doctrine  that  all  physical  action  is  mechanical  was 
stated  in  terms.  Even  during  the  life  of  Galilei — a 
year  before  his  death — Descartes  announced  that  "  all 
variations  of  matter,  or  all  diversity  of  its  forms,  de- 
pends on  motion."  *  And  nine  years  before  the 
appearance  of  Newton's  Principia  Thomas  Hobbes 
declared  that  "  change  (i.  e.,  physical  change)  is  of 
necessity  nothing  else  than  motion  of  the  parts  of  the 
body  changed,"  f  at  the  same  time  adding  that ."  there 
can  be  no  cause  of  motion  in  a  body  but  in  another 
body  contiguous  and  moved."  $  Leibnitz  was  even 
more  emphatic,  asserting  that  the  doctrine  in  question 
is  not  merely  an  experiential  induction,  but  a  self-evi- 

*  "  Omnis  materiae  variatio  sive  omnium  ejus  formarum  diversitas 
pendet  a  motu."  Cartes.  Princ.  Phil,  ii,  23. 

f  "  Necesse  est  ut  mutatio  aliud  non  sit  praeter  partium  corporis  mu- 
tati  motum."  Hobbes,  Philos.  prima,  pars  secunda,  ix,  9. 

\  "  Causa  motus  nulla  esse  potest  in  corporc  nisi  contiguo  et  moto." 
Ib. 


INTPvODUCTORY.  17 

dent  truth.  "  Everything  in  nature,"  he  said,  "  is  ef- 
fected mechanically — a  principle  which  can  be  made 
certain  by  reason  alone,  and  never  by  experiments, 
however  numerous  they  may  be."  *  He,  too,  insisted 
that  all  motion  is  caused  by  impact.  "  A  body  is  never 
moved  naturally,  except  by  another  body  which  presses 
in  touching  it."  f  Similarly  Huygens,  the  great  con- 
temporary of  Leibnitz  and  Newton,  said  that  "  in  true 
philosophy  the  causes  of  all  natural  effects  are,  and  in 
his  judgment  must  be,  conceived  mechanically,  unless 
we  are  to  renounce  all  hope  of  understanding  anything 
in  physics."  J  And  in  the  first  comprehensive  treatise 
on  physics  ever  published,  that  of  Musschenbroek,  it  is 
put  forth  as  an  axiom  that  "  no  change  is  induced  in 
bodies  whose  cause  is  not  motion."  * 

The  most  definite  statement,  however,  of  the  prop- 
osition that  the  true  aim  and  object  of  all  physical 
science  is  a  reduction  of  the  phenomena  of  nature  to  a 
coherent  mechanical  system  is  found  in  the  scientific 

*  "  Tout  se  fait  mecaniquement  dans  la  nature,  principe  qu'on  peut 
rendre  certain  par  la  seule  raison  et  jamais  par  les  experiences,  quelque 
nombre  qu'on  en  fasse."     Leibnitz,  Nouveaux  Essais,  Opp.  ed.  Erdmann, 
p.  383. 

f  "  Un  corps  n'est  jamais  mu  naturellement  que  par  un  autre  corps 
qui  le  presse  en  le  touchant."  6me  lettre  a  Clarke,  Erdmann,  p.  767. 
Hence  Wolff,  the  dogmatic  expounder  of  the  Leibnitian  philosophy : 
"  Corpus  non  agit  in  alterum  nisi  dura  in  ipsum  impingit."  Wolff,  Cos- 
mologia  gen.,  129. 

^  "...  in  vera  philosophia,  in  qua  omnium  efrectuum  causae  conci- 
piuntur  per  rationes  mechanicas :  id  quod  meo  judicio  fieri  debet  ni«i 
velimus  omnem  spem  abjicere  aliquid  in  physicis  intelligandi."  Huge- 
nii  Opp.  reliqua,  Amst.,  1728,  vol.  i  (Tract,  de  lumine),  p.  2. 

*  "  Nulla  autem  corporibus  inducitur  mutatio,  cujus  causa  non  fuerit 
motus,  sive  excitatus,  sive  minutus,  aut  suffocatus ;    omne  enim  incre- 
mentum  vel  decrementum,  generatio,  corruptio,  vel  qualiscunque  alteratio, 
quae  in  corporibus  contingit,  a  motu  pendet."     P.  v.  Musschenbroek,  In- 
trod.  ad.  philos.  naturalem,  vol..i,  cap.  1,  §  18  (ed.  Patav.,  1768). 


18  CONCEPTS  OF  MODERN  PHYSICS. 

writings  published  during  the  second  half  of  the  pres- 
ent century,  since  the  discoveries  made  in  organic  chem- 
istry by  the  aid  of  the  atomic  theory,  the  revelations  of 
the  spectroscope,  the  establishment  of  the  doctrine  of 
the  conservation  of  energy,  and  the  promulgation  of 
the  mechanical  theory  of  heat  with  its  complement,  the 
kinetic  theory  of  gases.  Thus  Kirchhoff,  one  of  the 
founders  of  the  theory  of  spectral  analysis,  said  in 
1865:  "The  highest  object  at  which  the  natural  sci- 
ences are  constrained  to  aim,  but  which  they  will  never 
reach,  is  the  determination  of  the  forces  which  are 
present  in  nature,  and  of  the  state  of  matter  at  any 
given  moment — in  one  word,  the  reduction  of  all  the 
phenomena  of  nature  to  mechanics."*  To  the  same 
effect  Helmholtz,  in  his  inaugural  address  "delivered  be- 
fore the  meeting  of  the  association  of  physicians  and 
naturalists  at  Innspruck,  in  1869 :  "  The  object  of  the 
natural  sciences  is  to  find  the  motions  upon  which  all 
other  changes  are  based,  and  their  corresponding  mo- 
tive forces — to  resolve  themselves,  therefore,  into  me- 
chanics.'^ No  less  pointed  are  the  words  of  Clerk 
Maxwell :  "  When  a  physical  phenomenon,"  he  writes, 
"  can  be  completely  described  as  a  change  in  the  con- 
figuration and  motion  of  a  material  system,  the  dynami- 
cal explanation  of  that  phenomenon  is  said  to  be  com- 

*  "  Das  hoechste  Ziel,  welches  die  Naturwissenschaften  zu  erstreben 
haben,  aber  nicmals  erreichen  werden,  1st  die  Ermittelung  der  Kraefte, 
welche  in  der  Natur  vorhanden  sind  und  des  Zustandes  in  dem  die  Ma- 
tcrie  in  einem  Augenblick  sich  befindet,  mit  einem  Worte,  die  Zurueck- 
f uehrung  aller  Naturerscheinungen  auf  die  Mechanik."  Kirchhoff,  Uebcr 
das  Ziel  der  Naturwissenschaften.  Prorectoratsrede,  Heidelberg,  1865. 
S.  9,  24. 

f  "  Das  Endziel  der  Naturwissenschaften  ist,  die  alien  andern  Veraen- 
derungen  zu  Grunde  liegenden  Bewegungcn  und  deren  Triebkraefte  zu 
finden,  also  sich  in  Mechanik  aufzuloesen."  Helmholtz,  Populaerwig- 
senschaftliche  Vortraegc,  i,  93. 


INTRODUCTORY.  19 

plete.  We  can  not  conceive  any  further  explanation 
to  be  either  necessary,  desirable,  or  possible,  for  as  soon 
as  we  know  what  is  meant  by  the  words  configuration, 
mass  and  force,  we  see  that  the  ideas  which  they  repre- 
sent •  are  so  elementary  that  they  can  not  be  explained 
by  means  of  anything  else."  * 

Citations  like  these,  from  the  writings  of  eminent 
physicists,  might  be  multiplied  almost  indefinitely. 
And,  if  we  turn  from  the  physicists  to  the  physiologists, 
we  meet  with  declarations  equally  explicit.  "  Every 
analysis,"  said  Ludwig  in  1852,  "  of  the  animal  organism 
has  thus  far  brought  to  light  a  limited  number  of  chem- 
ical atoms,  the  presence  of  the  light-  (heat-)  bearing 
aether  and  of  the  electric  fluids.  These  data  lead  to 
the  inference  that  all  the  phenomena  of  animal  life 
are  consequences  of  the  simple  attractions  and  repul- 
sions resulting  from  the  concurrence  of  these  element- 
ary substances."  f  In  a  similar  strain  Wundt,  writing 
twenty-five  years  later :  "  The  view  that  has  now  be- 
come dominant  (in  physiology),  and  is  ordinarily  desig- 
nated as  the  mechanical  or  physical  view,  has  its  origin 
in  the  causal  conception  long  prevalent  in  the  kindred 
departments  of  natural  science,  which  regards  nature  as 
a  single  chain  of  causes  and  effects  wherein  the  ulti- 
mate laws  of  causal  action  are  the  laws  of  mechanics. 

*  Clerk  Maxwell,  "  On  the  Dynamical  Evidence  of  the  Molecular  Con- 
stitution of  Bodies."  "  Nature,"  March  4  and  11,  1875. 

f  "  So  oft  nun  eine  Zergliederung  der  Icist-ungserzengenden  Einrich- 
tungen  des  thierischen  Koerpers  geschah,  so  oft  stiess  man  schliesslich 
auf  eine  begrenzte  Zahl  chemisher  Atome,  die  Gegenwart  des  Licht- 
(Waerme-)  Aethers  und  diejenige  der  electrischen  Fluessigkeiten.  Die- 
ser  Erfahrung  entsprechend  zieht  man  den  Schluss,  dass  alle  vom  thicr- 
ishen  Koerper  ausgehenden  Ersheinungen  eine  Folge  der  einfachen  An- 
ziehungen  und  Abstossungen  sein  moechten,  welche  an  jenen  elementaren 
Wesen  bei  einem  Zusammentreffen  derselben  beobachtet  werden."  Lud- 
wig, Lehrbuch  der  Physiologic  des  Menschen,  Band  i,  Einleitung,  p.  2. 


20  CONCEPTS  OF  MODERN  PHYSICS. 

Physiology  thus  appears  as  a  branch  of  applied  phys- 
ics, its  problem  being  a  reduction  of  vital  phenomena 
to  general  physical  laws,  and  thus  ultimately  to  the 
fundamental  laws  of  mechanics."  *  Still  more  broadly, 
Haeckel:  "The  general  theory  of  evolution  ...  as- 
sumes that  in  nature  there  is  a  great,  unital,  continuous 
and  everlasting  process  of  development,  and  that  all 
natural  phenomena  without  exception,  from  the  motion 
of  the  celestial  bodies  and  the  fall  of  the  rolling  stone 
up  to  the  growth  of  the  plant  and  the  consciousness  of 
man,  are  subject  to  the  same  great  law  of  causation — 
that  they  are  ultimately  to  be  reduced  to  atomic  me- 
chanics." f  This  theory,  Haeckel  declares,  "  is  the  only 
scientific  theory  which  affords  a  rational  explanation 
of  the  universe,  and  satisfies  the  craving  of  the  intellect 
for  causal  connections,  inasmuch  as  it  links  all-  the  phe- 
nomena of  nature  as  parts  of  a  great  unital  process  of 
development  and  as  a  S3ries  of  mechanical  causes  and 

*  "  Die  jctzt  zur  Herrschaft  gelangte  Auffassung  dagegen,  die  man  als 
die  physikalishe  oder  mechanistische  zu  bezeichnen  pflegt,  ist  aus  der  in 
den  verwandten  Zweigen  der  Naturwissenschaft  schon  laenger  zur  Gel- 
tung  gckommenen  causalen  Naturansicht  entsprungen,  welche  die  Natur 
als  einen  einzigen  Zusammenhang  von  Ursachen  und  Wirkungen  ansieht, 
wobei  als  letzte  Gesetze,  nach  denen  die  natuerlichen  Ursachen  wirken, 
sich  stets  die  Grundgesetze  der  Mechanik  ergeben.  Die  Pbysiologie  er- 
sbeint  daber  als  ein  Zweig  der  angewandten  Naturlehre.  Ihre  Aufgabe 
erkennt  sie  darin,  die  Lebenserscbeinungen  auf  die  allgemeinen  Natur- 
gesetze,  also  schliesslich  auf  die  Grundgesetze  des  Mechanik,  zurueckzu- 
fuehren."  Wundt,  Lchrbuch  der  Physiologic  des  Menschen,  4te  Au- 
flage,  p.  2. 

f  "Die  allgemeine  Entwickelungslchre  .  .  .  nimmt  an,  dass  in  der 
ganzen  Natur  ein  grosser,  einheitlicher,  ununterbrochener  und  ewigcr 
Entwickelungsvorgang  stattfindet,  und  dass  alle  Naturerscheinungen  ohne 
Ausnahme,  von  der  Bewegung  der  Himmelskoerper  und  dem  Fall  des 
rollenden  Steins  bis  zum  Wachsen  der  Pflanze  und  zum  Bewusstsein  des 
Menschen,  nach  einem  und  demselben  grossen  Causal-Gesetze  erfolgen, 
dass  alle  schliesslich  auf  Mechanik  der  Atome  zurueckzufuehren  sind." 
Haeckel,  Freie  Wissenschaft  und  freie  Lehre,  pp.  9,  10. 


INTRODUCTORY.  21  + 

effects."  *  In  the  same  sense  Huxley  speaks  of  "  that 
purely  mechanical  view  toward  which  modern  physi- 
ology is  striving."  f 

A  very  lucid  and  thorough  exposition  of  the  aims 
of  modern  physical  science  is  contained  in  the  following 
passage  taken  from  a  recent  lecture  of  Emil  Du  Bois- 
Reymond — equally  distinguished  as  a  physicist  and 
physiologist :  "  Natural  science — more  accurately  ex- 
pressed, scientific  cognition  of  nature,  or  cognition  of 
the  material  world  by  the  aid  and  in  the  sense  of  the- 
oretical physical  science — is  a  reduction  of  the  changes 
in  the  material  world  to  motions  of  atoms  caused  by 
central  forces  independent  of  time  or  a  resolution  of 
the  phenomena  of  nature  into  atomic  mechanics.  It  is 
a  fact  of  psychological  experience  that,  whenever  such 
a  reduction  is  successfully  effected,  our  craving  for 
causality  is,  for  the  time  being,  wholly  satisfied.  The 
propositions  of  mechanics  are  reducible  to  mathematical 
form,  and  carry  within  them  the  same  apodictic  cer- 
tainty which  belongs  to  the  propositions  of  mathemat- 
ics. When  the  changes  in  the  material  world  have  been 
reduced  to  a  constant  sum  of  potential  and  kinetic 
energy  inherent  in  a  constant  mass  of  matter,  there  is 
nothing  left  in  these  changes  for  explanation. 

"  The  assertion  of  Kant,  in  the  preface  to  the  <  Met- 
aphysical Rudiments  of  Natural  Science,'  that  '  in  every 
department  of  physical  science  there  is  only  so  much 
science,  properly  so  called,  as  there  is  mathematics,'  is 

*  "  Der  Monismus,  die  universale  Entwickelungstheorie,  oder  die  mo- 
nistische  Progenesistheorie  ist  die  einzige  wissenschaftliche  Theorie, 
welche  das  Weltganze  vernunftgemass  erklaert,  und  das  Causalitaets- 
beduerfniss  unserer  menschlichen  Vernunft  befriedigt,  indem  sie  alle 
Natur-Erscheinungen  als  Theile  eines  einheitlichen  grossen  Entwickelungs- 
Processes  in  mechanischen  Causal-Zusammenhang  bringt."  Ibid.,  p.  11. 

f  Lay  Sermons,  Addresses  and  Reviews  (Appletons'  ed.),  p.  331. 


22  CONCEPTS  OF  MODERN  PHYSICS. 

to  be  sharpened  by  substituting  <  mechanics  of  atoms ' 
for  f  mathematics.'  This  was  evidently  his  own  mean- 
ing when  he  denied  the  name  '  science '  to  chemistry. 
It  is  not  a  little  remarkable  that  in  our  time  chemistry, 
since  it  has  been  constrained,  by  the  discovery  of  sub- 
stitution, to  abandon  the  old  electro-chemical  dualism, 
has  seemingly  taken  a  retrograde  step  in  its  advance 
toward  science  in  this  sense.  The  resolution  of  all 
changes  in  the  material  world  into  motions  of  atoms 
caused  by  their  constant  central  forces  would  be  the 
completion  of  natural  science"  * 

*  "  Naturerkennen — genauer  gesagt,  naturwissenschaftliches  Erken- 
nen  oder  Erkennen  der  Koerperwelt  mit  Huelfe  und  im  Sinne  der  theo- 
retischen  Naturwissenschaft — ist  Zurueckf  uehren  der  Veraenderungen  in 
der  Koerperwelt  auf  Bewegungen  von  Atomen  die  durch  deren  von  der 
Zeit  unabhaengige  Centralkraefte  bewirkt  werden,  odef  Aufloesung  der 
Naturvorgaenge  in  Mechanik  der  Atome.  Es  ist  psychologische  Erfahr- 
ungsthatsache,  dass  wo  solche  Aufloesung  gelingt,  unser  Causalitaetsbe- 
duerfniss  vorlaeufig  sich  befriedigt  fuehlt.  Die  Saetze  der  Mechanik  sind 
mathematisch  darstellbar,  und  tragen  in  sich  dieselbe  apodiktische  Gewiss- 
heit,  wie  die  Saetze  der  Mathematik.  Indem  die  Veraenderungen  in  der 
Koerperwelt  auf  eine  constante  Summe  potentieller  und  kinetischer  Ener- 
gie,  welche  einer  constanten  Menge  von  Materie  anhaftet,  zurueckgef uehrt 
werden,  bleibt  in  diesen  Veraenderungen  selber  nichts  zu  erklaeren  uebrig. 

"  Kant's  Behauptung  in  der  Vorrede  zu  den  '  Metaphysischen  An- 
fangsgvuenden  der  Naturwissenschaft,'  '  dass  in  jeder  besonderen  Na- 
turlehre  nur  so  viel  eigentliche  Wissenschaft  angetroffen  werden  koenne, 
als  darin  Mathematik  anzutreffen  sei,'  ist  also  vielmehr  noch  dahin  zu 
verschaerfen,  dass  fuer  Mathematik  Mechanik  der  Atome  gesetzt  wird. 
Sichtlich  diess  ineinte  er  selber  als  er  der  Chemie  den  Namen  einer  Wis- 
senschaft absprach,  und  sie  unter  die  Experimentallehren  verwies.  Es 
ist  nicht  wenig  merkwuerdig  dass  in  unserer  Zeit  die  Chemie  indem  sie 
durch  die  Entdeckung  der  Substitution  gezwungen  wurde  den  electro- 
chemischen  Dualismus  aufzugeben,  sich  von  dem  Ziel,  eine  Wissenschaft 
in  diesem  Sinne  zu  werden,  scheinbar  wieder  weiter  entfernt  hat.  Denken 
wir  uns  alle  Veraenderungen  in  der  Koerperwelt  in  Bewegungen  von 
Atomen  aufgeloest,  die  durch  dcren  constante  Centralkraefte  bewirkt 
werden,  so  waere  das  Weltall  naturwissenschaftlich  erkannt."  Emil  Du 
Bois-Reymond,  "  Ueber  die  Grenzen  des  Naturerkennens,"  p.  2  seq. 


INTRODUCTORY.  2& 

With  few  exceptions,  scientific  men  of  the  present 
day  hold  the  proposition,  that  all  physical  action  is  me- 
chanical, to  be  axiomatic,  if  not  in  the  sense  of  being 
self-evident,  at  least  in  the  sense  of  being  an  induction 
from  all  past  scientific  experience.  And  they  deem 
the  validity  of  the  mechanical  explanation  of  the  phe- 
nomena of  nature  to  be,  not  only  unquestionable,  but  ab- 
solute, exclusive,  and  final.  They  believe  that  this 
validity  is  not  conditioned,  either  by  the  present  state 
of  human  intelligence,  or  by  the  nature  and  extent  of 
the  phenomena  which  present  themselves  as  objects  of 
scientific  investigation.  Thoughtful  men  like  Du  Bois- 
Reymond  have  at  times  suggested  that  it  is  not  unlim- 
ited ;  but  the  only  limits  assigned  to  it  are  those  of  the 
general  capacity  of  the  human  intellect.  Although 
they  concede  that  there  is  a  class  of  phenomena— those 
of  organic  life  —  which,  under  their  characteristic  as- 
pect, are  wholly  irreducible  by  the  mere  aid  of  mechan- 
ical principles,  it  is,  nevertheless,  insisted  that  these 
principles  constitute  the  only  intellectual  solvent  that 
can  be  applied  to  them,  and  that  the  residue  which  re- 
sists the  solution  is  to  be  relegated  for  ever  to  that  end- 
less array  of  facts  which  are  proof  against  all  the  re- 
agents of  scientific  cognition.  It  is  claimed  that,  if  it  is 
impossible  theoretically  to  construct  a  living  organism 
out  of  molecules  or  atoms,  and  mechanical  forces  under 
the  guidance  of  the  principle  of  the  conservation  of  en- 
ergy, the  laws  of  electric  or  magnetic  coercion,  the  first 
and  second  laws  of  thermo-dynamics,  etc.,  the  attempt 
to  frame  a  theory  of  life  in  harmony  with  the  laws  con- 
trolling ordinary  material  action  must  be  utterly  aban- 
doned. Such  a  claim  ought  notf  in  my  judgment,  to 
be  admitted  without  a  careful  examination  of  the 
grounds  upon  which  it  is  made.  It  is  my  purpose, 


24  CONCEPTS  OF  MODERN  PHYSICS. 

therefore,  in  the  following  pages  to  inquire  whether  or 
not  the  validity  of  the  mechanical  theory  of  the  uni- 
verse in  its  present  form,  and  with  its  ordinary  assump- 
tions, is  indeed  absolute  within  the  bounds  of  human 
intelligence,  and  to  this  end,  if  possible,  to  ascertain  the 
nature  of  this  theory  as  well  as  its  logical  and  psycho- 
logical origin.  Obviously  the  first  question  presenting 
itself  in  the  course  of  an  examination  into  its  validity 
is  whether  it  is  consistent  with  itself  and  with  the  facts 
for  the  explanation  of  which  it  is  propounded.  Our 
initial  problem,  then,  will  be  that  of  finding  an  answer 
to  this  question. 


/  V 

"UNIVERSITY; 


CHAPTEE  II. 

FIRST   PRINCIPLES     OF     THE   MECHANICAL    THEORY    OF   THE 
UNIVERSE. 

THE  mechanical  theory  of  the  universe  undertakes 
to  account  for  all  physical  phenomena  by  describing 
them  as  variances  in  the  structure  or  configuration  of 
material  systems.  It  strives  to  apprehend  all  phenom- 
enal diversities  in  the  material  world  as  varieties  in  the 
grouping  of  primordial  units  of  mass,  to  recognize  all 
phenomenal  changes  as  movements  of  unchangeable 
elements,  and  thus  to  exhibit  all  apparent  qualitative 
heterogeneity  as  mere  quantitative  difference.  In  the 
light  of  this  theory  the  ultimates  of  scientific  analysis 
are  mass  *  and  motion,  which  are  assumed  to  be  essen- 
tially disparate.  Mass,  it  is  said,  exists  independently 
of  motion  and  is  indifferent  to  it.  It  is  the  same 
whether  it  be  in  motion  or  at  rest.  Motion  may  be 
transferred  from  one  mass  to  another  without  destroy- 
ing the  identity  of  either. 

The  prime  postulate  of  all  science  is  that  there  is 
some  constant  amid  all  phenomenal  variations.  Science 
is  possible  only  on  *  the  hypothesis  that  all  change  is  in 
its  nature  transformation.  "Without  this  hypothesis  it 

*  It  is  hardly  necessary  to  say  that  I  purposely  designate  moss,  and 
not  (as  is  usual)  matter,  as  the  correlate  of  motion.  When  a  body  is  di- 
vested, in  thought,  of  all  those  qualities  which,  according  to  the  teachings 
of  modern  science,  are  in  their  nature  phases  of  motion,  the  residue  is 
not  matter,  but  mass. 


26  CONCEPTS  OF  MODERN  PHYSICS. 

could  discharge  neither  of  its  two  great  functions — those 
of  determining,  from  the  present  state  of  things,  the 
past  on  the  one  hand  and  the  future  on  the  other,  by 
exhibiting  the  one  as  its  necessary  antecedent,  and  the 
other  as  its  equally  necessary  consequent.  It  is  evident 
that  the  computations  of  science  would  be  utterly  frus- 
trated by  the  sudden  disappearance  of  one  or  more  of 
its  elements,  or  the  unbidden  intrusion  of  new  elements. 
If,  therefore,  scientific  analysis  yields  mass  and  motion 
as  its  absolutely  irreducible  elementary  terms — if  these 
terms  underlie  all  possible  transformations — it  follows 
that  both  are  quantitatively  invariable.  Accordingly 
the  mechanical  theory  of  the  universe  postulates  the 
conservation  of  both  mass  and  motion.  Mass  may  be 
transformed  by  an  aggregation  or  segregation  of  parts ; 
but  amid  all  these  transformations  it  persistently  remains 
the  same.  Similarly  motion  may  be  distributed  among 
a  greater  or  less  number  of  units  of  mass ;  it  may  be 
transferred  from  one  unit  of  mass  to  any  number  of 
units,  its  velocity  being  reduced  in  proportion  to  the 
number  of  units  to  which  the  transference  takes  place  ; 
nevertheless  the  sum  of  the  motions  of  the  several  units 
is  always  equal  to  the  motion  of  the  single  unit.  It 
may  be  changed  in  direction  and  form  ;  rectilinear  mo- 
tion may  become  curvilinear,  translatory  motion  may 
be  broken  up  into  vibratory  motion,  molar  motion  may 
be  converted  into  molecular  agitation ;  yet,  during  all 
these  changes,  it  is  never  increased,  diminished,  or  lost. 
The  conservation  of  mass  (or,  as  it  is  generally  but  in- 
accurately termed,  the  conservation  or  indestructibility 
of  matter)  has  long  been  a  standing  axiom  of  physical 
science.  The  conservation  of  motion  (i.  e.,  the  conser- 
vation of  energy,  which,  as  will  hereafter  appear,  is,  ac- 
cording to  the  mechanical  theory,  the  same  thing),  though 


GENERAL  PRINCIPLES  OF  MECHANICAL  THEORY.      27 

but  recently  formulated  as  a  distinct  scientific  principle, 
is  now  universally  regarded  as  of  equal  evidence  and 
axiomatic  dignity  with  its  older  counterpart.  Indeed, 
while  chemistry  is  said  to  be  founded  on  the  conserva- 
tion of  matter,*  the  recent  progress  of  theoretical  phys- 
ics has  consisted  mainly  in  the  solution  of  the  problem 
of  reconstituting  it  on  the  basis  of  the  conservation  of 
energy.  The  science  of  physics,  in  addition  to  the  gen- 
eral laws  of  dynamics  and  their  application  to  the  in- 
teraction of  solid,  liquid,  and  gaseous  bodies,  embraces 
the  theory  of  those  agents  which  were  formerly  desig- 
nated as  imponderables — light,  heat,  electricity,  mag- 
netism., etc.  ;  and  all  these  are  now  treated  as  forms  of 
motion,  as  different  manifestations  of  the  same  funda- 
mental energy,  and  as  controlled  by  laws  which  are 
simple  corollaries  from  the  law  of  its  conservation. 
The  only  apparent  exception  is  the  second  law  of  ther- 
modynamics, a  reduction  of  which,  however,  to  the 
principle  of  least  action,  or  rather  Hamilton's  extension 
of  it,  the  principle  of  varying  action,  has  been  attempted 
by  Boltzmann  and  Clausius,  while  others  (among  them 
Eankine,  Szily,  and  Eddy)  have  sought  to  derive  it  di- 
rectly from  the  principle  of  the  conservation  of  energy.. , 

It  is  thus  seen  that  the  theory  according  to  which 
the  cause  of  all  phenomenal  change  and  variety  in  na- 
ture is  motion,  and  all  apparent  qualitative  diversity  is 
in  reality  quantitative  difference,  involves  three  propo- 
sitions, which  may  be  stated  as  follows  : 

I.  The  primary  elements  of  all  natural  phenomena 

*  It  is  gradually  coming  to  be  understood  that  the  conservation  of 
energy  is  as  important  a  principle  in  chemistry  as  that  of  the  conserva- 
tion of  mass ;  but  as  yet  chemical  notation  takes  account  of  masses  only 
and  makes  no  exhibition  of  the  quantities  of  energy  gained  or  lost  in  any 
given  chemical  transformation.  •, 


28  CONCEPTS  OF  MODERN  PHYSICS. 

— the  ultimates  of  scientific  analysis — are  mass  and 
motion. 

II.  Mass  and  motion  are  disparate.     Mass  is  in- 
different to  motion,  which  may  be  imparted  to  it,  and 
of  which  it  may  be  divested,  by  a  transference  of  motion 
from  one  mass  to  another.     Mass  remains  the  same, 
whether  at  rest  or  in  motion. 

III.  Both  mass  and  motion  are  constant. 
Among  the  corollaries  from  the  first  and  second  of 

these  propositions  there  are  two  which  are  as  obvious 
as  they  are  important :  the  inertia  and  the  homogeneity 
of  mass.  Mass  and  motion  being  radically  disparate, 
it  is  evident  that  mass  can  not  be  motion  or  the  cause 
of  motion — it  is  inert.  And  mass  in  itself  can  not  be 
heterogeneous,  for  heterogeneity  is  difference,  and  all 
difference  is  caused  by  motion. 

The  propositions  above  set  forth  lie  at  the  base  of 
the  whole  mechanical  theory.  They  command  uni- 
versal assent  among  physicists  of  the  present  day,  and 
are  to  be  regarded  as  the  fundamental  axioms  of  mod- 
ern physical  science.  In  addition  to  these  propositions, 
however,  there  is  the  assumption,  generally  prevalent 
among  physicists  and  chemists,  of  the  molecular  or 
atomic  constitution  of  bodies,  according  to  which  mass 
is  not  continuous,  but  discrete,  being  an  aggregate  of 
unchangeable,  and,  in  that  sense  at  least,  simple  units. 
This  assumption  leads  to  four  other  propositions,  which, 
in  conjunction  with  the  principle  of  the  conservation  of 
both  mass  and  motion,  may  be  said  to  constitute  the 
foundations  of  the  atomo-mechanical  theory.  They  are 
these : 

1.  The  elementary  units  of  mass,  being  simple,  are 
in  all  respects  equal.  This  is  manifestly  nothing  more 
than  an  assertion  of  the  homogeneity  of  mass  in  con- 


GENERAL  PRINCIPLES  OF   MECHANICAL  THEORY.      29 

x 

formity  with  the  hypothesis  of  its  molecular  or  atomic 
constitution. 

2.  The  elementary  units  of  mass  are  absolutely  hard 
and  inelastic — a  necessary  consequence  of  their  sim- 
plicity, which  precludes  all  motion  of  parts,  and,  there- 
fore, all  change  of  figure. 

3.  The  elementary  units  of  mass  are  absolutely  in- 
ert and  therefore  purely  passive ;  hence  there  can  be 
no  mutual  action  between  them,  other  than  mutual  dis- 
placement caused  by  impulses  from  without. 

4.  All  potential  energy,  so  called,  is  in  reality  'ki- 
netic.    The  term  "  energy,"  in  the  language  of  modern 
physics,  denotes  the  cause  of  motion.     And  motion  can 
not  originate  in,  nor  can  it  be  converted  into,  anything 
but  motion.     The  invariable  units  of  mass  are  inert, 
whatever  be  their  position.     Energy  due  to  mere  posi- 
tion is  impossible. 

It  is  necessary  to  take  up  these  propositions  sever- 
ally in  their  order,  and  to  ascertain  whether,  and  to 
what  extent,  they  are  consistent  with,  and  serve  as  the 
explanation  of,  the  facts  of  scientific  experience. 


CHAPTER  III. 

THE   PROPOSITION  THAT   THE   ELEMENTARY  UNITS   OF   MASS 
AKE   EQUAL. 

IF  all  the  diversities  in  nature  are  caused  by  motion, 
it  follows  that  mass,  the  substratum  of  this  motion,  is 
fundamentally  homogeneous.  This  is  so  evident  that, 
in  the  first  distinct  announcements  of  the  mechanical 
theory,  the  two  propositions — the  principle  and  its  cor- 
ollary— appeared  side  by  side.  Thus  the  statement  of 
Descartes  cited  in  the  first  chapter  *  is  accompanied  by 
the  declaration  that  "  the  matter  which  exists  in  the 
world  is  everywhere  one  and  the  same."  f  It  is  true 
that  Descartes  did  not  assert  the  absolute  equality  of 
single  material  elements,  because  he  recognized  but  two 
primaiy  properties  of  matter,  extension  and  mobility, 
and  therefore  denied  its  atomic  constitution.  But,  when 
in  time  the  hypothesis  of  the  atomic  or  molecular 
structure  of  matter  became  one  of  the  cardinal  doctrines 
of  modern  physical  science,  the  postulate  of  the  funda- 
mental homogeneity  of  mass  necessarily  assumed  the 
form  of  an  assertion  of  the  absolute  equality  of  its  primor- 
dial units.  For  reasons  to  be  discussed  presently,  physi- 
cists, and  especially  chemists,  of  our  day  evince  a  disposi- 
tion to  ignore  this  essential  feature  of  the  mechanical 

*  Supra,  p.  16. 

•j-  "  Materia  itaque  in  toto  universo  una  et  eadem  existit."  Cart.  Princ. 
Phil.,  ii,  23. 


EQUALITY   OF   ELEMENTS  OF  MASS.  31 

theory ;  but,  among  those  who  understand  that  all  scien- 
tific theories  must  at  last  be  brought  to  the  test  of  con- 
sistency, it  has  rarely  failed  to  meet  with  direct  or  im- 
plied recognition.  "  Chemistry,"  says  Professor  Wundt, 
"  still  refers  the  divergent  qualities  of  matter  to  an  origi- 
nal qualitative  difference  between  the  atoms.  But  the 
whole  tendency  of  physical  atomism  is  to  derive  all  the 
qualitative  properties  of  matter  from  the  forms  of 
atomic  motion.  Thus  the  atoms  themselves  remain  as 
elements  utterly  devoid  of  quality"  *  Of  the  same  im- 
port are  the  words  of  Herbert  Spencer :  "  The  proper- 
ties of  the  different  elements  result  from  differences  of 
arrangement,  arising  by  the  compounding  and  recom- 
pounding  of  ultimate  homogeneous  units  "  ^  Even  in 
the  writings  of  distinguished  chemists  there  is  no  lack  of 
utterances  bearing  testimony  to  the  pressure  of  the  logi- 
cal necessity  which  constrains  the  modern  physicist  to 
insist  upon  the  fundamental  equality  of  the  material 
elements.  "  It  is  conceivable,"  says  Thomas  Graham, 
"  that  the  various  kinds  of  matter  now  recognized  as 
different  elementary  substances  may  possess  one  and 
the  same  ultimate  or  atomic  molecule  existing  in  differ- 
ent conditions  of  movement.  The  essential  unity  of 
matter  is  an  hypothesis  in  harmony  with  the  equal  action 
of  gravity  upon  all  bodies.  We  know  the  anxiety  with 
which  this  point  was  investigated  by  Newton  and  the 
care  he  took  to  ascertain  that  every  kind  of  substance, 

*  Die  abweichenden  Eigenschaften  der  Materie  verlegt  die  Chemie 
noch  jetzt  in  eine  urspruengliche  qualitative  Verachiedenheit  der  Atome'. 
Nun  geht  offenbar  die  ganze  Entwickelung  der  physikalischen  Atomistik 
darauf  aus,  alle  qualitativen  Eigenschaften  der  Materie  aus  den  Beweg- 
ungsformen  der  Atome  abzuleiten.  Die  Atome  selbst  bleiben  so  noth- 
wendig  als  volkommen  qualitaetslose  Elemente  zurueck.  "  Die  Theorie 
der  Materie,"  Deutsche  Rundschau,  December,  1875,  p.  381. 

f  Contemporary  Review,  June,  1872. 


32  CONCEPTS  OF  MODERN   PHYSICS. 

<  metals,  stones,  woods,  grain,  salts,  animal  substances,' 
etc.,  are  similarly  accelerated  in  falling,  and  are  there- 
fore equally  heavy. 

u  In  the  condition  of  gas,  matter  is  deprived  of 
numerous  and  varying  properties  with  which  it  ap- 
pears invested  when  in  the  form  of  a  liquid  or  solid. 
The  gas  exhibits  only  a  few  grand  and  simple  features. 
These  again  may  all  be  dependent  upon  atomic  or  mo- 
lecular mobility.  Let  us  imagine  one  kind  of  substance 
only  to  exist — ponderable  matter ;  and  further,  that  mat- 
ter is  divisible  into  ultimate  atoms,  uniform  in  size 
and  weight.  We  shall  then  have  one  substance  and  a 
common  atom.  With  the  atom  at  rest  the  uniformity 
of  matter  would  be  perfect.  But  the  atom  possesses 
always  more  or  less  motion,  due,  it  must  be  assumed, 
to  a  primordial  impulse.  This  motion  gives  rise  to 
volume.  The  more  rapid  the  movement  the  greater 
the  space  occupied  by  the  atom,  somewhat  as  the  orbit 
of  a  planet  widens  with  the  degree  of  projectile  veloci- 
ty. Matter  is  thus  made  to  differ  only  in  being  lighter 
or  denser  matter.  The  specific  motion  of  an  atom 
being  inalienable,  light  matter  is  no  longer  convertible 
into  heavy  matter.  In  short,  matter  of  different  density 
forms  different  substances — different  inconvertible  ele- 
ments as  they  have  been  considered. 

"  But,  further,  these  more  or  less  mobile,  or  light 
and  heavy  forms  of  matter,  have  a  singular  relation 
connected  with  equality  of  volume.  Equal  volumes 
of  two  of  them  can  coalesce  together,  unite  their  move- 
ment, and  form  a  new  atomic  group,  retaining  the 
whole,  the  half,  or  some  simple  proportion  of  the 
original  movement  and  consequent  volume.  This  is 
chemical  combination.  It  is  directly  an  affair  of  vol- 
ume, and  only  indirectly  connected  with  weight. 


EQUALITY  OF  ELEMENTS  OF  MASS.        33 

Combining  weights  are  different,  because  the  densi- 
ties, atomic  and  molecular,  are  different."  * 

Views  analogous  to  those  of  Graham  are  held  by  C. 
R.  A.  Wright,  who  suggests  "  that  there  is  but  one  kind 
of  primordial  matter,  all  so-called  elements  and  com- 
pounds being,  as  it  were,  allotropic  modifications  of  this 
matter,  differing  from  one  another  in  the  amount  of 
energy  latent  per  unit  of  mass."f  And  although  Prout's 
conjecture,  that  the  several  chemical  elements  are  really 
compounds  or  allotropic  forms  of  hydrogen,  has  been 
definitively  abandoned  (even  by  Dumas  and  others  who 
at  divers  times  sought  to  revive  it),  it  having  been 
shown  that  the  hypothesis,  according  to  which  the 
atomic  weights  of  all  the  elements  are  exact  multiples 
of  that  of  hydrogen,  is  untenable,  yet  attention  has  late- 
ly been  drawn  to  the  fact  that  there  seem  to  be  spec- 
troscopic  indications  of  the  predominance  of  a  few  gase- 
ous elements,  such  as  hydrogen  and  nitrogen,  in  certain 
nebulae  which  appear  to  represent  the  earlier  stages  of 
planetary  or  stellar  development,  and  of  a  gradual  in- 
crease of  metallic  and  other  substances  in  more  ad- 
vanced forms — in  other  words,  of  a  progressive  differen- 
tiation of  matter,  a  gradual  advance  from  homogeneity 
to  heterogeneity,  on  the  successive  stages  of  planetary 
or  stellar  evolution.^ 

Now,  while  the  absolute  equality  of  the  primordial 
units  of  mass  is  thus  an  essential  part  of  the  very  foun- 
dations of  the  mechanical  theory,  the  whole  modern 
science  of  chemistry  is  based  upon  a  principle  directly 

*  "  Speculative  Ideas  respecting  the  Constitution  of  Matter,"  Phil. 
Mag.,  4th  ser.,  vol.  xxvii,  p.  81  seq. 

f  Chemical  News,  October  31,  1873. 

|  Cf.  F.  W.  Clarke,  "  Evolution  and  the  Spectroscope,"  Popular  Sci- 
ence Monthly,  January,  1873,  p.  320  seq.  Lockyer's  recent  investiga- 
tions have  brought  these  views  into  great  prominence. 


34  CONCEPTS  OF  MODERN  PHYSICS. 

subversive  of  it — a  principle  of  which  it  has  recently  been 
said  that  "  it  holds  the  same  place  in  chemistry  that  the 
law  of  gravitation  does  in  astronomy."  *  This  principle 
is  known  as  the  law  of  Avogadro  or  Ampere.  It  im- 
ports that  equal  volumes  of  all  substances,  when  in  the 
gaseous  state  and  under  like  conditions  of  pressure  and 
temperature,  contain  the  same  number  of  molecules — 
whence  it  follows  that  the  weights  of  the  molecules  are 
proportional  to  the  specific  gravities  of  the  gases ;  that, 
therefore,  these  being  different,  the  weights  of  the 
molecules  are  different  also ;  and,  inasmuch  as  the  mole- 
cules of  certain  elementary  substances  are  monatomic 
(i.  e.,  consist  of  but  one  atom  each),  while  the  molecules 
of  various  other  substances  contain  the  same  number  of 
atoms,  that  the  ultimate  atoms  of  such  substances  are  of 
different  weights.- 

The  law  of  Avogadro,  though,  like  all  physical 
theories,  an  hypothesis,  is  believed  to  be  the  only  hy- 
pothesis which  is  competent  to  account  for  the  well- 
known  variation  of  the  volume  of  a  gas  inversely  as 
the  pressure  (law  of  Boyle  or  Mariotte)  and  directly  as 
the  absolute  temperature  (law  of  Charles)  as  well  as 
for  the  combination  of  gases  in  simple  volumetric  pro- 
portions (law  of  Gay-Lussac) ;  and  it  has  served  as  the 
basis  of  innumerable  deductions  respecting  the  forma- 
tion and  transformation  of  chemical  compounds  which 
have  thus  far  met  with  unfailing  experimental  verifica- 
tion. 

That  this  cardinal  principle  of  modern  theoretical 
chemistry  is  in  utter  and  irreconcilable  conflict  with  the 
first  proposition  of  the  atomo-mechanical  theory  is  ap- 
parent at  a  glance.  No  reconciliation,  certainly,  is  pos- 
sible on  the  hypothesis  suggested  by  Graham.  For 

*  J.  P.  Cooke,  The  New  Chemistry,  p.  13. 


EQUALITY   OF  ELEMENTS  OF   MASS.  35 

that  accounts  for  differences  of  density  by  attributing 
to  equal  primordial  atoms  unequal  volumes  resulting 
from  their  occupancy  of  unequal  spaces  by  virtue  of 
differences  in  the  velocities  of  movement  with  which 
the  several  kinds  of  atoms  are  supposed  to  be  inalien- 
ably endowed.  It  accounts  for  inequalities  in  the  vol- 
umes of  equal  masses,  not  for  inequalities  of  mass  in 
equal  volumes,  and  can  not  serve  as  an  explanation  of 
the  latter,  unless  it  is  supplemented  by  the  further  as- 
sumption— -to  which,  indeed,  it  lends  little,  if  any,  aid 
— that  some,  if  not  all,  of  the  molecules  are  compounds 
or  aggregates  of  different  degrees  of  complexity.  Two 
masses  or  molecules  of  equal  volumes  can  be  of  different 
densities  or  weights  only  if  the  number  of  units  con- 
tained in  one  is  different  from  the  number  of  units  in 
the  other.  But  Avogadro's  law  constrains  the  chemist 
to  assume  that  the  molecules  of  various  elementary  sub- 
stances, notwithstanding  the  diversity  of  their  weights, 
consist  of  the  same  number  of  atoms.  Thus  hydrogen 
and  chlorine,  whose  molecular  weights  are  two  and 
seventy-one  respectively,  are  both  held  to  be  diatomic, 
i.  e.,  their  molecules  are  held  to  consist  of  two  atoms 
each.  In  the  case  of  monads,  or  univalent  elements, 
such  as  those  just  mentioned,  the  reasoning  upon  which 
this  assumption  rests  is  very  simple.  One  volume  of 
hydrogen  combines  with  one  volume  of  chlorine,  form- 
ing two  volumes  of  hydrochloric  acid.  Each  volume  of 
this  compound,  according  to  Avogadro's  law,  contains 
as  many  molecules  as  either  volume  of  the  constituent 
simple  elements  before  combination ;  the  two  volumes 
of  the  compound,  therefore,  contain  twice  as  many 
molecules  as  either  volume  of  the  constituents.  But, 
in  each  molecule  of  the  compound,  both  hydrogen  and 
chlorine  are  present,  whence  it  follows  that  each  mole- 


36  CONCEPTS  OF  MODERN  PHYSICS. 

cule  of  hydrogen,  as  well  as  each  molecule  of  chlorine, 
must  have  contributed  at  least  one  atom  to  each  mole- 
cule of  hydrochloric  acid,  and  thus  must  have  consisted 
of  at  least  two  atoms. 

The  argument  in  the  case  of  dyads  (such  as  oxygen, 
sulphur,  selenium,  etc.),  and  other  elements  of  still 
higher  quanti  valence,  though  somewhat  less  simple,  is 
equally  cogent  upon  the  basis  of  Avogadro's  law. 

It  may  be  said  that  the  law  in  question  determines 
only  the  minimum  number  of  atoms  in  each  molecule, 
leaving  the  maximum  indeterminate,  so  that,  after  all, 
the  molecule  of  greater  weight  may  be  of  correspond- 
ingly greater  complexity.  But  here  we  encounter  an 
obstacle  presented  by  a  branch  of  the  atomic  theory  in 
physics — the  science  of  thermo-dynamics.  Modern  sci- 
ence regards  heat  as  a  form  of  energy — as  consisting  in 
an  agitation  of  the  molecules  or  atoms  whereof  bodies 
are  composed  ;  and,  in  the  case  of  gaseous  bodies  at 
least,  it  discriminates  between  that  part  of  this  energy 
which  is  exhibited  in  the  form  of  temperature,  attribut- 
ing it  to  translatory  motions  of  the  molecules,  or  rather 
of  their  centers  of  mass,  and  another  part — the  internal 
energy,  so  called — which  is  supposed  to  be  dependent 
upon  oscillatory  or  rotatory  motions  of  their  component 
atoms.  It  has  been  shown,  experimentally,  that  the 
ratio  of  the  specific  heat  of  a  gas  at  constant  pressure 
to  that  at  constant  volume*  falls  short  of  the  value 
assigned  to  it  by  the  theory  upon  the  supposition  that 
all  the  heat  imparted  to  a  gaseous  body  is  expended  in 
producing  a  translatory  motion  of  the  molecules,  the 

*  The  "  specific  heat "  (i.  e.  the  heat  required  to  raise  the  tempera- 
ture of  a  unit  of  mass  of  any  substance  one  degree)  of  a  gas  at  constant 
pressure  under  which  it  expands,  is  necessarily  greater  than  that  at  con- 
stant volume,  because  in  the  former  case  part  of  the  heat  is  expended  in 
the  mechanical  work  of  expansion. 


EQUALITY   OF  ELEMENTS   OF   MASS.  37 

effect  being  expansion,  or  increased  pressure,  or  both ; 
and  this  difference  is  accounted  for  by  the  assumption 
that  part  of  the  heat  is  converted  into  intramolecular 
agitation,  i.  e.,  into  motions  of  the  particles  within  the 
molecule  which  do  not  affect  its  position  or  action  as  a 
whole.  ISTow,  it  is  readily  seen  and  has  been  shown  by 
Clausius,  Boltzmann,  Maxwell,  and  others,  that  the  en- 
ergy thus  converted  into  intramolecular  or  interatomic 
agitation  must  increase  as  the  complexity  of  the  molec- 
ular constitution  increases ;  it  would  become  enormous, 
therefore,  if  a  molecule  consisted  of  a  number  of  atoms 
so  great  as  to  be  sufficient  to  account  for  the  differences 
between  the  molecular  weights  of  the  elements.  The 
molecular  weight  of  chlorine,  for  example,  is  35'5  times 
as  great  as  that  of  hydrogen ;  and  if  these  weights  are 
in  proportion  to  the  number  of  atoms  contained  in  each 
molecule,  it  becomes  necessary  to  assume — even  grant- 
ing that  hydrogen  is  strictly  diatomic — that  each  chlo- 
rine molecule  is  composed  of  no  less  than  seventy-one 
atoms.  But,  if  this  assumption  were  valid,  nearly  all 
the  heat  imparted  to  chlorine  would  be  absorbed,  i.  e., 
converted  into  internal  energy,  and  its  calculated  spe- 
cific heat  would  far  exceed  the  amount  ascertained  by 
actual  experiment. 

There  are  thus  difficulties  not  of  a  speculative,  but 
of  a  purely  physical  and  chemical  nature,  which  render 
the  indefinite  multiplication  of  atoms  within  the  mole- 
cule, so  as  to  account  for  the  diversity  of  molecular 
weights,  wholly  inadmissible.  Several  elementary  sub- 
stances are  known  to  conform  to  Avogadro's  law  only 
on  the  supposition  that  they  are  monatomic.  Among 
them  is  mercury,  whose  molecular  weight  coincides 
with  its  atomic  weight  as  established  by  all  the  chemi- 
cal tests  applicable  to  it,  including  that  of  Dulong  and 


38  CONCEPTS  OF  MODERN  PHYSICS. 

Petit's  law.  And  it  has  been  demonstrated  by  Kundt 
and  Warburg  *  that  the  ratio  of  the  specific  heat  of 
mercurial  vapor  at  constant  pressure  to  that  at  constant 
volume,  as  ascertained  by  experiment,  is  precisely  equal 
to  its  value  calculated  upon  the  basis  of  the  absolute 
simplicity  of  the  mercurial  molecule  and  of  the  non- 
absorption  of  any  part  of  the  heat  in  intramolecular 
action. 

In  view  of  all  this  there  seems  to  be  no  escape  from 
the  conclusion  that  the  claim,  according  to  which  mod- 
ern physical  science  is  throughout  a  partial  and  pro- 
gressive solution  of  the  problem  of  reducing  all  physi- 
cal phenomena  to  a  system  of  atomic  mechanics,  is  very 
imperfectly,  if  at  all,  countenanced  by  the  actual  con- 
stitution of  theoretical  chemistry — that  this  science, 
which  is  peculiarly  conversant  about  atoms  and  their 
motions,  is  founded  upon  propositions  destructive  of 
the  very  basis  upon  which  alone  a  consistent  super- 
structure of  atomic  mechanics  can  be  reared.  And 
there  appears  to  be  little  ground  for  the  hope  that 
these  propositions  may  be  speedily  abandoned ;  for,  in 
the  opinion  of  the  most  distinguished  chemists  of  the 
day,  such  an  abandonment  would  throw  the  mass  of 
chemical  facts,  laboriously  ascertained  by  experiment 
and  observation  (induced,  partly  at  least,  by  the  propo- 
sitions in  question)  into  a  state  of  hopeless  prescientific 
confusion. 

In  reference  to  the  speculations  of  those  who  seek 
to  deduce  the  specific  differences  between  the  ultimate 
units  of  mass  from  differences  between  their  supposed 
inalienable  velocities  of  motion  or  amounts  of  latent 
energy,  it  is  to  be  said,  not  only  that  they  fail  to  afford 
a  solution  of  the  difficulties  of  theoretical  chemistry  in 

*  Pogg.  Ann.,  vol.  clvii,  p.  353. 


EQUALITY  OF  ELEMENTS  OF  MASS.  39 

the  presence  of  the  inexorable  demands  of  the  mechan- 
ical theory,  but  also  that  the  attribution  of  inalienable 
energy  or  motion  to  a  given  mass  is  repugnant  to  the 
fundamental  postulate  of  the  absolute  indifference  of 
mass  to  motion.  Helmholtz  and  others  have  investi- 
gated the  conditions  of  vortex  motion  in  a  perfectly 
homogeneous,  incompressible  and  frictionless  fluid, 
which  (as  Maxwell  has  shown)  is  of  necessity  continu- 
ous and  can  not  be  molecular  or  atomic.  If  these  con- 
ditions could  be  realized,  we  should  have  constant  but 
undistinguishable  volumes  of  a  permanently  homoge- 
neous fluid,  so  called,  endowed  with  constant  quantities 
of  inalienable  motion.  /;33ut  no  energy  or  motion  can 
inhere  essentially  in  distinct  and  separate  masses  (mole- 
cules or  atoms)  if,  as  the  mechanical  theory  assumes, 
mass  and  motion  are  disparate — if  mass  is  indifferent 
to  motion  so  as  to  remain  the  same  whether  in  motion 
or  at  rest,  and  if  motion  is  transferable  from  one  mass 
to  another.  This  is  one  of  the  points  distinctly  insisted 
upon  by  Sir  Isaac  Newton,  the  greatest  among  the 
founders  of  the  mechanical  theory.  Newton  distin- 
guishes between  two  kinds  of  force — the  force  of  iner- 
tia (vis  inertiae\  and  impressed  force  (vis  impressa). 
The  former  alone  according  to  him  is  vis  insita,  i.  e., 
inheres  in  matter ;  while  of  the  latter  he  expressly  says 
that  "  this  force  consists  in  action  alone  and  does  not 
abide  in  the  body  after  action."  *  ^ 

*  "  Consistit  haec  vis  in  actione  sold,  neque  post  actionem  permanet  in 
corpore."  Phil.  Nat.  Princ.  Math.,  def.  iv  (ed.  Le  Seur  et  Jacquier,  vol. 
i,  p.  4). 


CHAPTER  IV. 

THE   PROPOSITION   THAT   THE   ELEMENTARY  UNITS    OF   MASS 
ABE   ABSOLUTELY    HARD   AND   INELASTIC. 

FROM  the  essential  disparity  of  mass  and  motion  and 
the  simplicity  of  the  elementary  units  of  mass  it  follows 
that  these  units  are  perfectly  hard  and  inelastic.  Elas- 
ticity involves  motion  of  parts  and  can  not,  therefore, 
be  an  attribute  of  truly  simple  atoms.  "  The  concept 
'  elastic  atom,'  "  justly  observes  Professor  Wittwer,  "  is 
a  contradiction  in  terms,  because  elasticity  presupposes 
parts  the  distances  between  which  can  be  increased  and 
diminished."  * 

The  early  founders  of  the  mechanical  theory  re- 
garded the  absolute  hardness  of  the  component  par- 
ticles of  matter  as  an  essential  feature  of  the  original 
order  of  nature.  "  It  seems  probable  to  me,"  says  Sir 
Isaac  Newton,  "  that  God  in  the  beginning  formed  mat- 
ter in  solid,  massy,  hard,  impenetrable,  movable  parti- 
cles of  such  sizes  and  figures,  and  with  such  other  prop- 
erties and  in  such  proportion  to  space  as  most  conduced 
to  the  end  for  which  he  formed  them ;  and  that  these 
primitive  particles  being  solids  are  incomparably  harder 
than  any  porous  bodies  compounded  of  them ;  even  so 

*  "  Der  Begriff  '  elastisches  Atom '  ist  erne  cojitradictio  in  adjectis,  da 
die  Elasticitaet  immer  wieder  Theile  voraussetzt,  die  sich  einander  naeh- 
ern,  die  sich  von  einander  entfernen  koennen."  Beitraege  zur  Molecu- 
larphysik,  Schloemilch's  Zeitschrift  fuer  Math,  und  Phys.,  vol.  xv,p.  114. 


RIGIDITY   OF  ELEMENTS  OF  MASS.  41 

very  hard  as  never  to  wear  or  break  in  pieces ;  no  or- 
dinary power  being  able  to  divide  what  God  himself 
made  one  in  the  first  creation."  * 

Strangely  enough,  while  the  requirement,  by  the 
mechanical  theory,  of  the  absolute  rigidity  of  the  ele- 
mentary units  of  mass  is  no  less  imperative  than  that 
of  their  absolute  simplicity,  it  meets  with  an  equally 
signal  denial  in  modern  physics.  The  most  conspicuous 
among  the  hypotheses  which  have  been  devised  since 
the  general  adoption  of  the  modern  theories  of  heat, 
light,  electricity  and  magnetism,  and  the  establishment 
of  the  doctrine  of  the  conservation  of  energy,  in  order 
to  afford  consistent  ground  for  the  mechanical  inter- 
pretation of  physical  phenomena,  is  that  known  as  the 
c  kinetic  theory  of  gases.  In  the  light  of  this  theory  a  '• 
gaseous  body  is  a  swarm  of  innumerable  solid  particles 
incessantly  moving  about  with  different  velocities  in 
rectilinear  paths  of  all  conceivable  directions,  the  veloci- 
ties and  directions  being  changed  by  mutual  encounters 
at  intervals  which  are  short  in  comparison  with  ordi- 
nary standards  of  duration,  but  indefinitely  long  as  com- 
pared with  the  duration  of  the  encounters.  It  is  readi- 
ly seen  that  these  motions  would  soon  come  to  an  end 
if  the  particles  were  wholly  inelastic,  or  imperfectly 
elastic.  For  in  that  case  there  would  be  loss  of  motion  | 
at  every  encounter.  The  assumed  perpetuity  of  the 
motion  of  the  particles,  therefore,  leads  to  the  necessity 
of  asserting  their  perfect  elasticity.  And  this  neces- 
sity results,  not  merely  from  the  peculiar  exigencies  of 
the  kinetic  theory  of  gases,  but  also  from  the  principle 
of  the  conservation  of  energy  in  its  general  application 
to  the  ultimate  constituents  of  sensible  masses,  if  these 
constituents  are  supposed  to  be  in  motion.  In  the  case 

*  Opticks,  fourth  ed.,  p.  375. 


4:2  CONCEPTS   OF  MODERN  PHYSICS. 

of  the  collision  of  ordinary  inelastic  or  partially  elastic 
bodies  there  is  a  loss  of  motion  which  is  accounted  for 
by  the  conversion  of  the  motion  thus  lost  into  an  agita- 
tion of  the  minute  parts  composing  the  colliding  bodies. 
But  in  atoms  or  molecules  destitute  of  parts  no  such 
conversion  is  possible,  and  hence  we  are  constrained  to 
assume  that  the  ultimate  molecules  of  a  gaseous  body 
are  absolutely  elastic. 

The  necessity  of  attributing  perfect  elasticity  to  the 
elementary  molecules  or  atoms  in  view  of  the  kinetic 
theory  of  gases  has  been  expressly  recognized  by  all  its 
founders.  "  Gases,"  says  Kroenig,*  "consist  of  atoms 
which  behave  like  solid,  perfectly  elastic  spheres  mov- 
ing with  definite  velocities  in  void  space."  This  state- 
ment is  adopted  by  Clausius  f  and  emphasized  by  Max- 
well, the  first  part  of  whose  essay,  "  Illustration  of  the 
Dynamical  Theory  of  Gases,"  is  a  treatise  "  on  the  mo- 
tions and  collisions  of  perfectly  elastic  spheres."  \  And 
the  highest  scientific  authorities  are  equally  explicit  in 
declaring  that  the  hypothesis  of  the  atomic  or  molecular 
constitution  of  matter  is  in  conflict  with  the  doctrine  of 
the  conservation  of  energy,  unless  the  atoms  or  mole- 
cules are  assumed  to  be  perfectly  elastic.  "We  are 
forbidden,"  says  Sir^William  Thomson,*  "  by  the  mod- 
ern theory  of  the  conservation  of  energy  to  assume  in- 
elasticity or  anything  short  of  perfect  elasticity  of  the 
ultimate  molecules,  whether  of  ultra-mundane  or  mun- 
xdane  matter." 

Naturally,  eminent  advocates  of  the  kinetic  hy- 
pothesis have  taxed  their  ingenuity  in  the  search  of 

*  Pogg.  Ann.,  vol.  xcix,  p.  316. 
f  lb.,  vol.  c,  p.  353. 

\  Phil/Mag.,  4th  ser.,  vol.  xix,  p.  19. 

*  Ib.,  vol.  xlv,  p.  321. 


RIGIDITY   OF  ELEMENTS  OF  MASS.  43  , 

methods  for  the  extrication  of  the  mechanical  theory, 
from  the  dilemma  in  which  it  is  thus  involved.  The 
most  notable  effort  thus  far  made  is  that  of  Sir  "William 
Thomson,  in  the  form  of  a  conjecture  suggested  by  the 
researches  of  Helmholtz,*  respecting  the  properties  of 
rotational  motion  in  an  absolutely  homogeneous,  in- 
compressible, perfect  fluid,  to  which  reference  has  al- 
ready been  made  in  the  preceding  chapter.  Thomson 
imagines  the  omnipresence  of  this  fluid,  and  supposes 
that  atoms  are  in  fact^yortex-rings  formed  by  rotational 
movements  within  it.  Such  rings  would  be  permanent,\ 
of  invariable  volume  due  to  an  invariable  quantity  of 
motion,  though  susceptible  of  a  great  variety  of  form ; 
and  some  of  their  features,  such  as  their  modes  of  im- 
plication, would  be  indestructible ;  they  would  be  ca- 
pable of  being  knotted  on  themselves  or  linked  with 
other  vortex-rings,  but  could  never  be  unknotted  or 
untied ;  finally  they  would  be  incapable  of  interpenetra- 
tion  or  coalescence,  and  their  mutual  approaches  would 
result  in  rebounds  similar  to  the  resilience  of  perfectly 
elastic  bodies. 

While  we  willingly  yield  our  homage  to  the  sagaci- 
ty displayed  in  this  attempt  to  relieve  the  mechanical 
theory  from  one  of  its  most  fatal  embarrassments,  it  is 
to  be  feared  that  its  success  is  altogether  illusory.  For, 
it  seems  to  be  evident  that  motion  in  a  perfectly  homo- 
geneous, incompressible  and  therefore  continuous  fluid 
is  not  sensible  motion.  All  partition  of  such  a  fluid  is  \ 
purely  ideal ;  in  spite  of  the  displacement  of  any  por- 
tion of  it  by  another  portion,  a  given  space  would  at 
any  moment  present  the  same  quantity  of  substance 
absolutely  indistinguishable  from  that  present  there  a 

*  Cf.  Crelle-Borchardt's  Journal  fuer  reine  und  angewandte  Mathe- 
matik,  vol.  Iv,  p.  25. 


44  CONCEPTS  OF  MODERN  PHYSICS. 

moment  before.  There  would  be  no  phenomenal  dif- 
ference or  change.  A  fluid  both  destitute  and  incapable 
of  difference  is  as  impossible  a  vehicle  of  real  motion 
as  pure  space ;  it  is  as  useless  for  the  purpose  of  ac- 
counting for  the  phenomena  of  material  action-  as  the 
quasi-material  medium  without  inertia  of  which  Roger 
v  Cotes  said  that  it  was  not  to  be  distinguished  from  a 
vacuum.* 

Again,  as  Maxwell  has  observed,f  the  vortex-ring 
atoms  moving  in  the  hypothetical  fluid  would  lack  the 
essential  attribute  of  matter :  inertia.  Such  atoms  would 
consist,  not  in  the  substance  of  the  omnipresent  fluid, 
but  simply  in  the  motions  induced  therein.  Of  these 
motions  the  persistence  of  both  mass  and  energy  would 
have  to  be  predicated,  and  from  them  the  concretions 
of  mass,  together  with  all  the  phenomena  exhibited  by 
sensible  matter,  would  have  to  be  derived.  But  that  is 
impossible.  From  its  very  nature  motion  can  not  be  the 
bearer  of  motion,  nor  can  it,  by  itself,  be  the  generator 
of  momentum  which  is  essentially  the  product  of  two 
antagonistic  factors,  and  which  would  be  utterly  extin- 
guished by  the  suppression  of  either.  Upon  the  basis 
of  the  mechanical  theory,  the  fundamental  antithesis 
between  mass  and  motion,  inertia  and  energy,  can  not  be 
destroyed  without  an  obliteration  of  all  the  distinctions 
which  constitute  the  elements  of  our  conceptions  re- 
specting the  nature  of  physical  action. 

Another  attempt,  somewhat  analogous  to  that  of  Sir 
William  Thomson,  to  dispense  with  the  necessity  of  en- 

*  "  Qui  coelos  materi&  fluida  repletos  esse  volunt,  hanc  vero  non  in- 
ertem  esse  statuunt,  hi  verbis  tollunt  vacuum,  re  ponunt.  Nam  cum 
hujusmodi  materia  fluida  ratione  nulla  secerni  possit  ab  inani  spatio; 
disputatio  tota  fit  de  rerum  nominibus,  non  de  naturis.  Praef.  in  New- 
toni  Phil.  Nat.  Princ.  Math.,  ed.  Le  Seur  &  Jacquier,  p.  25. 

f  Encycl.  Brit.,  ninth  ed.,  s.  v.  Atom. 


RIGIDITY   OF  ELEMENTS   OF  MASS.  45' 

dowing  the  elementary  atoms  with  the  intrinsic  prop- 
erty of  elasticity  has  been  made  by  A.  Seechi.  This 
distinguished  physicist  and  astronomer  also  derives  the 
resilience  of  the  ultimate  particles  from  their  rotatory 
motion  ;  but  his  atoms,  unlike  those  of  Thomson,  are  real 
corpuscles  separated  by  wide  interstitial  spaces,  and  not 
mere  movements  in  a  continuous  and  incompressible 
sethereal  medium.  Secchi  clearly  apprehends  the  inad- 
missibility  of  attributing  elasticity  to  simple  elementary 
atoms.  "It  is  evident,"  he  says,*  "that,  while  it  is\ 
possible  to  admit  its  existence  in  a  compound  molecule, 
the  same  thing  can  not  be  done  in  the  case  of  elementary 
atoms.  Indeed,  elasticity  in  the  received  sense  pre- 
supposes void  spaces  in  the  interior  of  the  molecule 
whose  form  is  changed  by  compression  so  as  to  return, 
afterward,  to  its  original  figure.  Now,  we  regard  the 
atoms  as  impenetrable,  and  not  as  groups  of  solid  par- 
ticles ;  hence  they  can  not  include  void  spaces  which 
permit  their  dilatation  and  contraction. 

"  In  truth,  what  we  call  a  molecule  of  a  simple  (i.  e., 
chemically  undecomposable)  gas  is  not  an  elementary 
atom,  or  at  least  is  not  necessarily  one.  Inasmuch  as 
this  gaseous  molecule  is  an  aggregate  of  veritable  atoms, 
it  may  well  be  that  it  has  internal  pores,  and,  generally, 
a  number  of  properties  which  do  not  belong  to  its  con- 
stituent atoms ;  it  is  not  absurd,  therefore,  to  suppose 
it  to  be  endowed  with  elasticity.  JIuygens  has  ad- 
mitted this  hypothesis  for  the  sether.  In  his  opinion 
the  sethereal  particles  are  composed  of  smaller  ones ; 
but  on  closer  examination  it  is  seen  that  this  is  a  mere 
shifting  of  the  difficulty,  and  not  a  solution  of  it.  "We 
hope  to  be  able  to  show  that  it  is  nowise  necessary  to 
accept  such  an  elasticity  as  a  primitive  force,  and  that 

*  L'unitS  des  forces  physiques,  2me  6d.,  p.  47  seq. 
3 


46  CONCEPTS  OF  MODERN  PHYSICS. 

the  apparent  repulsion  of  the  atoms  and  their  reciprocal 
collisions  can  be  simply  referred  to  an  appropriate  mo- 
tion, it  being  sufficient  for  this  purpose  to  suppose  them 
to  be  in  rotation.  Let  us  prove  this : 

"  Among  the  beautiful  theorems  discovered  by  Poin- 
sot  respecting  the  impact  of  bodies  in  rotation  is  found 
one  relating  to  their  reflection  from  a  resisting  obstacle. 
It  teaches  us  that  by  virtue  of  its  rotation  alone  a  hard 
and  inelastic  body  can  rebound  absolutely  like  a  body 
^perfectly  elastic  ;  more  than  that :  one  of  these  bodies, 
thrown  against  a  fixed  obstacle,  is  often  sent  back  with 
a  velocity  superior  to  its  initial  velocity.  The  profound 
mathematician  shows  how  this_phenomenon,  paradoxi- 
cal  as  it  seems,  is  due  to  the  /transformation  of  part  of 
its  rotatory  motion  into  motion  of  translation  \  whence 
results  an  increase  of  the  velocity  of  the  center  of  grav- 
ity. According  to  the  ordinary  theories  of  impact,  in 
which  no  account  is  taken  of  the  motion  of  rotation, 
the  preceding  proposition  is  absurd,  and  nevertheless  it 
is  perfectly  established.  Thus,  by  the  side  of  cases  of 
ordinary  reflection  we  find  the  phenomena  of  progres- 
sion •  we  might  also,  using  the  expression  of  Poinsot, 
call  them  <n$gative  reflections*. 

"  In  negative  reflection  after  impact,  the  center  of 
gravity  of  the  body  returns  with  a  velocity  superior  to 
that  which  it  had  at  first.  These  questions  form  a 
wholly  new  and  very  interesting  branch  of  mechanics  ; 
they  are  easily  demonstrated  by  compounding  the  two 
movements  of  rotation  and  translation,  considered  with 
reference  to  the  centers  of  gravity,  of  rotation  and  of 
percussion;  and  we  readily  understand  that  generally 
it  may  be  said :  an  impact,  whatever  it  may  be,  can 
never  simultaneously  annihilate  in  a  body  the  two  mo- 
tions of  rotation  and  of  translation ;  \f or,  when  the  im- 


RIGIDITY   OF  ELEMENTS   OF  MASS.  £ 

pact  is  eccentric,  it  can  destroy  rotation  and  not  trans- 
lation, and,  when  the  direction  of  the  impact  passes 
through  thex  center  of  gravity,  it  can  annihilate  transla- 
tion, but  not  rotation.  Thus,  the  quantity  of  motion 
lost  on  the  one  side  is  gained  on  the  other ;  the  rotation 
may  either  be  reversed  or  simply  accelerated,  according 
to  the  point  of  the  body  which  is  struck ;  whence  the 
notion  of  centers  of  conversion.  Examples  of  reflection 
succeeding  the  impact  of  bodies  in  rotation  are  found 
in  the  movements  of  disks  and  quoits,  the  impact  of 
spinning-tops,  etc.  Billiard-players  know  perfectly  how 
the  rotation  of  the  balls  modifies  the  laws  relating  to 
the  impact  of  elastic  bodies  as  established  in  the  ele- 
mentary treatises."  * 

Unfortunately,  the  theory  thus  advanced  finds  little 
support  in  Poinsot's  theorems.  Secchi  maintains  that 
the  impact  of  a  rotating  body,  when  it  is  eccentric, 
"  can  destroy  rotation,  but  not  translation,"  and,  when 
its  direction  passes  through  the  center  of  gravity,  "  it 
can  annihilate  translation,  but  not  rotation,"  so  that  in 
either  case  "  the^quantity  of  motion  lost  on  the  one  side 
is  gained  on  the  other."  f  But,  from  a  careful  examina- 
tion of  Poinsot's  memoir,  it  appears  that,  after  the  colli- 

*  The  theorems  to  which  Secchi  refers  are  contained  in  the  last  of  a 
series  of  memoirs  (Questions  Dynarniques  sur  la  Percussion  des  Corps) 
contributed  by  M.  Poinsot  to  Liouville's  Journal  de  Mathematiques 
pures  et  appliquees,  2me  serie,  t.  ii  (1857),  p.  281  seq.,  and  t.  iv  (1859), 
p.  421  seq.  This  remarkable  memoir  was  published  (and  probably  writ- 
ten) by  the  octogenarian  geometer  shortly  before  his  death  ;  the  last  in- 
stallment, indeed,  was  published  after  his  death  in  the  same  number  of 
Liouville's  Journal  which  contained  the  addresses  pronounced  at  his 
funeral  by  MM.  Bertrand  and  Mathieu. 

f  Secchi  invariably  speaks  of  loss  or  gain  of  "  quantity  of  motion  " ; 
but  his  argument  requires  that  this  should  be  interpreted  as  meaning  loss 
or  gain  of  energy.  Whether  or  not  this  is  his  own  meaning,  I  do  not 
undertake  to  say. 


48  CONCEPTS   OF  MODERN  PHYSICS. 

sion  of  rotating  inelastic  bodies,  tbeir  rotation,  or  trans- 
lation, or  both  are  conserved,  or  the  increase,  diminu- 
tion or  loss  of  the  one  is  compensated  for  by  the 
diminution,  increase  or  gain  of  the  other,  only  in  cer- 
tain special  cases.  Poinsot  shows  *  that,  when  a  rotat- 
ing inelastic  body  encounters  a  fixed  obstacle,  it  de- 
pends on  the  distance  between  the  spontaneous  center 
of  rotation  and  the  center  of  gravity  whether  the  body 
shall  be  reflected  with  a  translatory  velocity  greater 
than,  equal  to,  or  less  than  its  initial  velocity,  or  shall 
lose  its  translatory  velocity  altogether.  In  the  first 
place,  there  are  always,  between  the  center  of  gravity 
and  the  center  of  percussion,  "  two  points  such  that,  if 
the  rotating  body  strikes  the  obstacle  in  the  line  of 
either,  its  center  of  gravity  will  be  reflected  with  an 
increased  velocity."  f  In  the  second  place,  "  there  are 
always,  in  every  advancing  rotating  body,  two  points 
of  perfect  reflection,  i.  e.,  two  points  such  that,  if  the 
body  strikes  an  obstacle  in  the  line  of  either,  it  will  be  • 
reflected  with  a  velocity  perfectly  equal  to  the  velocity 
with  which  it  is  animated,":):  so  that  "the  center  of 
gravity  of  the  body  is  reflected  in  space  as  though  the' 
body  were  perfectly  elastic."  JSut,  when  this  occurs, 
the  body  loses,  in  the  one  case  one  third,  and  in  the 
other  two  thirds  of  its  velocity  of  rotation.®  Finally,,in 
the  third  case,  "  if  the  obstacle  is  presented,  either  to 
the  center  of  gravity  or  to  the  center  of  percussion,  the 
velocity  of  translation  is  equally  destroyed,  the  only 
difference  between  the  two  cases  being  that  in  the  first 
case  only  the  velocity  of  translation  is  destroyed  with- 
out alteration  of  the  velocity  of  rotation ;  while  in  the 

*  Liouville,  Journal,  etc.,  2me  serie,  t.  ii,  p.  288  seq. 
f  L.  c.,  p.  304.  \  L.  c.,  p.  305.  , 

*  L.  c.,  p.  307. 


RIGIDITY   OF  ELEMENTS  OF  MASS.  4.9 

second  case  both  the  velocity  of  translation  and  the 
velocity  of  rotation  are  annihilated."  * 

The  truth  is,  therefore,  that  in  the  only  instances  of 
perfect  reflection  specified  by  Poinsot  there  is  a  loss  of 
either  one  third  or  two  thirds  of  the  rotatory  motion 
not  compensated  for  by  any  increase  of  translatory  mo- 
tion, and  that  there  are  cases  of  impact  in  which  both 
the  motion  of  translation  and  that  of  rotation  simulta- 
neously disappear.f 

That  Secchi  should  have  deemed  it  possible  to  de- 
volve the  duty  of  conserving  the  energy  of  colliding 
atoms  upon  rotation  as  a  substitute  for  the  "  occult 
quality "  of  perfect  elasticity,,  seems  almost  incredible 
when  we  come  to  consider  the  use  he  makes  of  his  own 
theory.  This  theory,  according  to  him,  serves  as  an 
explanation  of  a  number  of  things,  among  which  are  the 
formation  of  molecular  aggregates  from  simple  atoms, 
and  the  phenomena  of  gravitation.  The  aggregation  of 
atoms,  so  as  to  form  compound  molecules,  he  explains 
thus :  $  "  Suppose  an  extreme  case,  viz.,  the  collision  of 

*  L.  c.,  p.  308. 

f  Although  I  have  long  since  become  utterly  indifferent  to  questions 
and  claims  of  priority,  it  may  not  be  improper  to  say  here  that  the  fore- 
going pages  were  written  before  I  had  seen  the  very  able  pamphlet  "  Das 
Raethsel  der  Schwerkraft "  (Braunschweig,  Vieweg  und  Sohn,  1879),  of 
D.  C.  Isenkrahe,  with  whom  I  am  happy  to  find  myself  in  accord  as  to  the 
validity  of  Secchi's  attempt  to  deduce  the  property  of  perfect  resilience 
from  the  rotation  of  inelastic  bodies  by  the  aid  of  Poinsot's  exposition  of 
the  theory  of  rotation,  although.I  can  not,  of  course,  accede  to  Isenkrahe's 
own  theory  of  gravitation.  There  are  other  coincidences — all  the  more 
interesting  because  they  are,  no  doubt,  wholly  accidental — between  the 
criticisms  contained  in  this  pamphlet  of  Spiller's  speculations  and  my  es- 
timate of  them  which  was  first  published  in  The  Popular  Science  Month- 
ly, January,  1874.  It  is  to  be  regretted  that  Isenkrahe,  before  publish- 
ing his  essay,  had  not  seen  William  B.  Taylor's  important  memoir,  herein- 
after referred  to,  on  "  Kinetic  Theories  of  Gravitation." 

\  L'unite",  etc.,  p.  51  seq. 


50  CONCEPTS  OF  MODERN  PHYSICS. 

two  atoms  endowed  solely  with  translation,  or,  again, 
impinging  on  one  another  so  that  they  can  not  rebound  " 
(which  would  happen  if  rotating  atoms  collided  in  the 
direction  of  their  axes  of  rotation).  "  Evidently  the 
atoms  will  remain  united  in  the  same  way  as  the  bodies 
called  '  hard  '  by  the  mechanicians,  and  they  will  form 
a  system  animated  by  the  movement  of  translation  re- 
sulting from  the  two  other  movements.  This  system 
will  be  able  to  act  like  a  single  corpuscule  whose  mass 
is  double,  triple  or  generally  a  multiple  of  that  of  a 
simple  atom  according  as  two  or  a  greater  number  of 
atoms  are  thus  united.  Here  we  have  an  obvious  in- 
stance of  an  aggregate  of  atoms  bound  to  each  other, 
not  by  the  influence  of  any  sort  of  attraction,  but  by 
simple  inertia."  Judging  from  this  passage,  Secchi 
could  hardly  have  been  ignorant  of  the  fact  that  the 
collision  of  rotating  inelastic  bodies  does  not  always  re- 
jsult  in  pseudo-elastic  resilience.  And  in  its  application 
to  the  phenomena  of  gravitation  his  theory  is  plainly 
destructive  of  its  own  foundations.  He  seeks  to  ac- 
count for  gravitation  upon  the  assumption  that  the 
density  of  the  sethereal  medium  which  surrounds  all 
ponderable  bodies  or  molecules  increases  from  their  cen- 
ters outward ;  *  and  this  increase  of  density  is  said  to  be 

*  This  supposition  is  identical  with  that  of  Sir  Isaac  Newton,  who 
in  his  letter  to  Boyle  (Newton's  Works,  ed.  Horsley,  vol.  iv,  p.  385  seq.), 
speculating  on  the  "  cause  of  gravity  "  said :  "  I  will  suppose  aether  to 
consist  of  parts  differing  from  one  another  in  subtil ty  by  infinite  degrees 
...  in  such  a  manner  that  from  the  top  of  the  air  to  the  surface  of  the 
earth,  and  again  from  the  surface  of  the  earth  to  the  centre  thereof,  the 
aether  is  insensibly  finer  and  finer.  Imagine  now  any  body  suspended  in 
the  air  or  lying  on  the  earth,  and  the  aether  being  by  the  hypothesis 
grosser  in  the  pores  which  are  in  the  upper  parts  of  the  body  than  in 
those  which  are  in  the  lower  parts,  and  that  grosser  aether  being  less  apt 
to  be  lodged  in  those  pores  than  the  finer  aether  below,  it  will  endeavor 
to  get  out  and  give  way  to  the  finer  aether  below,  which  can  not  be  with- 
out the  bodies  descending  to  make  room  above  for  it  to  go  into." 


RIGIDITY   OF  ELEMENTS  OF  MASS.  5V 

a  consequence  of  the  progressive  conversion  of  rotatory 
into  translatory  motion  of  the  sethereal  particles,  so  that 
these  particles  are  perpetually  driven  from  the  "  centers 
of  agitation  "  outward.  "  Evidently,"  says  Secchi,*  "  a 
center  of  agitation,  even  when  it  is  single,  provided  it  f 
is  animated  by  a  movement  sufficiently  energetic  and 
durable,  may  determine  the  agitation  of  an  unlimited 
medium,  and  so  modify  it  that  the  density,  least  at  its 
center,  increases  in  proportion  as  we  approach  the  cir- 
cumference." Secchi  assigns  no  reason  why  there  should 
be  a  perpetual  increase  in  the  translatory  motions  of  the 
sethereal  particles  at  the  expense  of  their  rotatory  mo- 
tions— why  the  transformation  should  always,  or  gen- 
erally, be  from  rotatory  into  translatory  motion  and  not 
conversely ;  nor  does  he  indicate  the  source  of  that 
"  energetic  and  durable  "  agitation  at  the  center  which 
is  said  to  be  productive  of  a  continual  agitation  of  a 
boundless  sethereal  sphere  ;  so  that  his  explanation  of  the 
phenomena  of  gravitation  is  of  very  questionable  valid- 
ity. But,  waiving  this ;  surely,  if  the  rotatory  motion 
of  the  hard  particles  is  gradually  transformed  into  trans- 
latory motion,  there  is,  by  his  own  showing,  an  ejxd.io 
their  resilience,  and  we  are  again  in  full  presence  of  the 
unsolved  problem  of  the  reconciliation  between  the  per- 
petual impact  of  simple,  hard,  and  therefore  inelastic 
atoms  and  the  conservation  of  their  initial  energy. 

The,  difficulty,  then,  appears  to  be  inherent  and  in- 
soluble/ There  is  no  method  known  to  physical  sci- 
ence which  enables  it  to  renounce  the  assumption  of  the 
perfect  elasticity  of  the  particles  whereof  ponderable 
bodies  and  their  hypothetical  imponderable  envelopes 
are  said  to  be  composed,  however  clearly  this  assump- 
tion conflicts  with  one  of  the  essential  requirements  of 
the  mechanical  theory. 

*  z.  c.,  p.  538. 


CHAPTEE  Y. 

THE   PROPOSITION    THAT   THE    ELEMENTARY    UNITS    OF  MASS 
ARE   ABSOLUTELY   INERT. 

MASS  and  motion  being  mutually  inconvertible,  mass 
is  absolutely  inert.  It  can  induce  motion  in  another 
mass  only  by  transferring  a  part  or  the  whole  of  its  own 
motion.  And,  inasmuch  as  motion  can  not  exist  by 
itself,  but  requires  mass  as  its  necessary  substratum, 
such  transference  can  not  take  place  unless  the  masses 
between  which  it  occurs  are  in  contact.  All  physical 
action,  therefore,  is  by  impact  ;  action  at  a  distance  is 
impossible;  there  are  in  nature  no  pulls,  but  only 
thrusts ;  and  all  force  is  not  merely  (in  the  language  of 
Newton)  vis  impressa,  but  vis  a  tergo. 

The  necessity  of  reducing  all  physical  action  to  im- 
pact has  been  a  persistent  tenet  among  physicists  ever 
since  the  birth  of  modern  physical  science.  And  yet, 
here  again,  as  in  the  cases  discussed  in  the  two  preced- 
ing chapters,  science  rises  in  revolt  against  its  own  fun- 
damental assumptions.  Its  first  and  greatest  achieve- 
ment was  Newton's  reduction  of  all  the  phenomena  of 
celestial  motion  to  the  principle  of  universal  gravita- 
tion— to  the  principle  that  all  bodies  whatever  attract 
each  other  with  a  force  proportional  directly  to  their 
masses  and  inversely  to  the  squares  of  the  distances  be- 
tween them. 

That  the  doctrine  of  universal  gravitation,  in  the 


INERTIA  OF  ELEMENTS  OF  MASS.  $ 

sense  of  an  attraction  at  a  distance  without  the  inter- 
vention of  a  medium  capable  of  propagating  mechanical 
impulses,  is  at  variance  with  the  elements  of  the  me- 
chanical theory  was  felt  by  no  one  more  distinctly  than 
by  Newton  himself.  At  the  very  outset  of  his  Prin- 
cipia  he  carefully  guarded  against  the  imputation  that 
he  looked  upon  gravity  as  an  essential  and  inherent 
attribute  of  matter  or  believed  the  mutual  attraction  of 
bodies  to  be  an  ultimate  physical  fact.  The  force  which 
urges  bodies  in  their  central  approach  was  to  him,  as 
he  expressly  says,  a  purely  mathematical  concept  in- 
volving no  consideration  of  real  and  primary  physical 
causes.*  And,  evidently  apprehensive  lest  this  dis- 
claimer should,  after  all,  be  lost  sight  of,  he  repeated  it, 
in  terms  no  less  explicit,  at  the  close  of  his  great  work. 
"  The  reason  of  these  properties  of  gravity,"  he  said, 
"  I  have  not,  as  yet,  been  able  to  deduce  ;  and  I  frame 
no  hypotheses."  f  If,  after  this,  there  were  still  room 
for  doubt  as  to  Newton's  opinions  respecting  the  nature 
of  gravity,  it  would  be  removed  by  the  well-known 
passage  in  his  third  letter  to  Bentley.  "  It  is  incon- 
ceivable that  inanimate  brute  matter  should,  without 
the  mediation  of  something  else  which  is  not  material, 
operate  upon  and  affect  other  matter,  without  mutual 
contact,  as  it  must  do  if  gravitation,  in  the  sense  of 
Epicurus,  be  essential  and  inherent  in  it.  And  this  is 
the  reason  why  I  desired  you  would  not  ascribe  innate 
gravity  to  me.  That  gravity  should  be  innate,  inherent 
and  essential  to  matter,  so  that  one  body  may  act  upon 

*  "  Mathematicus  duntaxat  est  hie  conceptus.  Nam  virium  causas  et 
scdes  physicas  jam  non  expcndo"  Princ.,  Def .  viii. 

f  "  Rationem  vero  harum  gravitatis  proprietatum  nondumpotui  dedu- 
cere  ;  et  hypotheses  non  fngo."  Princ.,  Schol.  Gen.  ad  fin.  The  same 
disclaimer  is  implied  in  the  words  of  a  scholium  to  the  29th  Theorem, 
Prop.  69,  Book  I,  of  the  Principia. 


54  CONCEPTS  OF   MODERN  PHYSICS. 

another  at  a  distance,  through  a  vacuum,  without  the 
mediation  of  anything  else  by  and  through  which  their 
action  may  be  conveyed  from  one  to  another,  is  to  me 
so  great  an  absurdity  that  I  believe  no  man,  who  has  in 
philosophical  matters  a  competent  faculty  of  thinking, 
can  ever  fall  into  it.  Gravity  must  be  caused  by  an 
agent  acting  constantly  according  to  certain  laws  ;  but, 
whether  this  agent  be  material  or  immaterial,  I  have 
left  to  the  consideration  of  my  readers."  * 

There  is  still  further  evidence  that  Newton  regarded 
universal  gravitation  as  a  secondary  phenomenon,  to  be 
explained  on  the  principles  of  ordinary  impact  or  press- 
ure. In  the  later  edition  of  his  Opticks  he  pro- 
pounds certain  "  Queries  "  relating  to  the  possibility  of 
deducing  some  of  the  properties  of  light  from  the  un- 
dulations of  an,  all-pervading  aether,  and  adds  (Query 
21) :  "  Is  not  this  medium  much  rarer  within  the  dense 
bodies  of  the  sun,  stars,  planets,  and  comets,  than  in  the 
empty  celestial  spaces  between  them  ?  And,  in  passing 
from  them  to  great  distances,  doth  it  not  grow  denser 
and  denser  perpetually,  and  thereby  cause  the  gravity 
of  those  great  bodies  towards  one  another,  and  of  their 
parts  towards  the  bodies ;  every  body  endeavoring  to 
go  from  the  denser  parts  of  the  medium  towards  the 
rarer."  f 

*  Newton's  Works,  ed.  S.  Horsley,  vol.  iv,  p.  438.  Zoellner  (Prin- 
cipien  einer  electrodynamischen  Theorie  der  Materie,  vol.  i,  preface)  at- 
tempts to  break  the  force  of  this  and  other  passages  in  the  writings  of 
Newton,  but,  as  it  appears  to  me,  wholly  without  avail. 

f  Opticks,  4th  ed.,  p.  325.  The  "  Queries  "  appeared  for  the  first 
time  in  the  second  edition  of  the  Opticks,  in  the  preface  to  which 
Newton  again  says  :  "  To  shew  that  I  do  not  take  gravity  for  an  essential 
property  of  bodies,  I  have  added  one  question  concerning  its  cause,  chus- 
ing  to  propose  it  by  way  of  a  question,  because  I  am  not  satisfied  about 
it  for  want  of  experiments."  I  have  already  cited  in  another  place  (su- 
pra, p.  42)  a  similar  exposition  of  his  views  in  the  letter  to  Boyle. 


INERTIA  OF  ELEMENTS  OF   MASS.  55^ 

Notwithstanding  these  explicit  declarations,  New- 
ton's contemporaries  took  alarm  at  the  apparent  return 
of  occult  causes  into  the  domain  of  physics.  It  is  in- 
teresting to  note  the  energy  with  which  the  philosophers 
and  mathematicians  of  his  day  protested  against  the 
assumption  of  physical  action  at  a  distance.  Huygens 
did  not  hesitate  to  say  that  "  Newton's  principle  of  at- 
traction appeared  to  him  absurd."  Leibnitz  called  it 
"an  incorporeal  and  inexplicable  power"  ;  John  Ber- 
noulli, who  sent  to  the  Academy  of  Paris  two  essays,  in 
which  he  sought  to  explain  the  movements  of  the  plan- 
ets by  an  improved  form  of  the  Cartesian  theory  of 
vortices,  denounced  "  the  two  suppositions  of  an  at- 
tractive faculty  and  a  perfect  void  "  as  "  revolting  to 
minds  accustomed  to  receiving  no  principle  in  physics 
save  those  which  are  incontestable  and%  evident."  Nor 
did  the  principle  of  distant  action  find  greater  favor 
with  the  physicists  and  astronomers  of  a  later  genera- 
tion. Euler  observed  that  the  action  of  gravity  must 
be  due  either  to  the  intervention  of  a  spirit  or  to  that 
of  some  subtle  material  medium  escaping  the  perception 
of  our  senses ;  and  he  insisted  that  the  latter  was  the 
only  admissible  alternative,  although  the  exact  demon- 
stration of  the  origin  of  gravitative  force  might  be 
difficult  or  impossible.*  His  great  rival  and  antagonist, 
D'Alembert,  relegated  gravity  to  that  class  of  causes 
productive  of  motion  whose  real  nature  is  to  us  entirely 
unknown,  in  contradistinction  to  action  by  impact,  of 
which  we  have  a  clear  mechanical  conception,  f  And, 

*  Euler,  "  Theoria  motus  corporum  solidorum,"  p.  68.  See  also  his 
"  Lettres  a  une  princesse  d'Allemagne,"  No.  68.  October  18,  1760. 

f  D'Alembert,  "  Dynamique  "  (2me  ed.),  p.  ix  seq.  It  is  well  known 
how  slowly  and  reluctantly  the  Newtonian  philosophy  found  recognition 
and  acceptance  in  France,  where  Cartesianisra  held  undisputed  sway  al- 
most to  the  end  of  the  eighteenth  century.  What  the  Cartesians  gener- 


56  CONCEPTS  OF  MODERN  PHYSICS. 

in  spite  of  the  assertion  of  John  Stuart  Mill  and  others 
that  the  thinkers  of  our  own  time  have  emancipated 
themselves  from  the  old  prejudice  against  actio  in  dis- 
tanSj  it  is  easy  to  show  that  it  is  almost,  if  not  quite,  as 
prevalent  now  as  it  was  two  centuries  ago.  To  cite  but 
a  few  instances :  Professor  Challis,  who  has  spent  a 
number  of  years  in  the  effort  to  establish  a  complete 
hydro-dynamical  theory  of  attraction,  says  :  "  There  is 
no  other  kind  of  force  than  pressure  by  contact  of  one 
body  with  another.  This  hypothesis  is  made  on  the 
principle  of  admitting  no  fundamental  ideas  that  are 
not  referable  to  sensation  and  experience.  It  is  true 
that  we  see  bodies  obeying  the  influence  of  an  external 
force,  as  when  a  body  descends  toward  the  earth  by  the 
action  of  gravity ;  so  far  as  the  sense  of  sight  informs 
us,  we  do  not  in  such  cases  perceive  either  the  contact 
or  the  pressure  of  another  body.  But  we  have  also 
the  sense  of  touch  or  of  pressure  by  contact — for  in- 
stance, of  the  hand  with  another  body — and  we  feel  in 
ourselves  the  power  of  causing  motion  by  such  press- 
ure. The  consciousness  of  this  power  and  the  sense 
of  touch  give  a  distinct  idea,  such  as  all  the  world  un- 
derstands and  acts  upon,  as  to  how  a  body  may  be 
moved ;  and  the  rule  of  philosophy  which  makes  per- 
sonal sensation  and  experience  the  basis  of  scientific 
knowledge,  as  they  are  the  basis  of  the  knowledge  that 

ally  thought  of  the  distant  action  of  gravitation  may  be  gathered  from  a 
paper  read  by  Saurin  to  the  Academic  des  Sciences  in  1709,  from  which 
Edleston  ("  Correspondence  between  Newton  and  Cotes,"  p.  213)  makes 
the  following  quotation :  "  H  (Newton)  airne  mieux  conside"rer  la  pesan- 
teur  comme  une  qualite"  inhe'rente  dans  les  corps  et  ramener  les  idees 
tant  decrie'es  de  qualite  occulte  et  d' attraction."  If  we  abandon  mechan- 
ical principles  (i.  e.,  the  principles  of  mechanical  impact  and  propulsion), 
he  continues,  "  nous  voila  replonges  de  nouveau  dans  les  anciennes  tene- 
bres  du  peripatetisme  dont  le  ciel  nous  veuille  preserver." 


INERTIA   OF  ELEMENTS  OF  MASS.  57 

regulates  the  common  transactions  of  life,  forbids  rec- 
ognizing any  other  mode  than  this.  When,  therefore, 
a  body  is  caused  to  move  without  apparent  contact  and 
pressure  of  another  body,  it  must  still  be  concluded  that 
the  pressing  body,  although  invisible,  exists,  unless  we 
are  prepared  to  admit  that  there  are  physical  operations 
which  are,  and  ever  will  be,  incomprehensible  by  us. 
This  admission  is  incompatible  with  the  principles  of 
the  philosophy  I  am  advocating,  which  assume  that  the 
information  of  the  senses  is  adequate,  with  the  aid  of 
mathematical  reasoning,  to  explain  phenomena  of  all 
kinds.  .  .  .  All  physical  force  being  pressure,  there 
must  be  a  medium  by  which  the  pressure  is  exerted."  * 
With  equal  vigor  the  "  assumption "  of  universal  at- 
traction is  reprobated  as  "  an  absurdity "  by  James 
Croll.  "  No  principle  "  he  contends,  "  will  ever  be 
generally  received  that  stands  in  opposition  to  the  old 
adage,  '  A  thing  can  not  act  where  it  is  not,'  any  more 
than  it  would  were  it  to  stand  in  opposition  to  that 
other  adage,  '  A  thing  can  not  act  before  it  is  or  when 
it  is  not.'  "  f  Secchi  protests  in  almost  the  same  words. 
"  We  have  said  elsewhere,"  he  declares,  "  how  impossi- 
ble it  is  to  conceive  what  is  called  an  attractive  force  in 
the  strict  sense  of  the  term,  that  is,  to  imagine  an  ac- 
tive principle  having  its  seat  within  the  molecules  and 
acting  without  a  medium  through  an  absolute  void. 
This  amounts  to  an  admission  that  bodies  act  upon  each 
other  at  a  distance,  that  is,  where  they  are  not :  an  ab- 
surd hypothesis — equally  absurd  in  the  case  of  enor- 
mous and  in  that  of  very  small  distances."  :f  Friedrich 

*  "  On  the  Fundamental  Ideas  of  Matter  and  Force  in  Theoretical 
Physics,"  Phil.  Mag.,  4th  series,  vol.  xxxi,  p.  467. 

f  "  On  Certain  Hypothetical  Elements  in  the  Theory  of  Gravitation," 
Phil.  Mag.,  4th  series,  vol.  xxxiv,  p.  460. 

\  L'unit6,  etc.,  p.  532  seq. 


58  CONCEPTS  OF  MODERN  PHYSICS. 

Mohr  (who  appears  to  be  entitled  to  the  honor  of  hav- 
ing distinctly  announced  the  principle  of  the  conser- 
vation of  energy,  even  before  Julius  Robert  Mayer) 
formulates  his  scientific  creed  in  a  number  of  "  Theses," 
among  which  is  this  :  "  Gravity  can  not  act  except  by 
the  interposition  of  ponderable  matter."  *  So  also  E. 
Du  Bois-Reymond  :  "  Forces  acting  through  void  space 
are  in  themselves  inconceivable,  nay  absurd,  and  have 
become  familiar  concepts  among  physicists  since  New- 
ton's time 'from  a  misapprehension  of  his  doctrine  and 
against  his  express  warning."  f  And  finally  Balfour 
Stewart  and  P.  G.  Tait :  "  Of  course,  the  assumption 
of  action  at  a  distance  may  be  made  to  account  for  any- 
thing ;  but  it  is  impossible  (as  Newton  long  ago  pointed 
out  in  his  celebrated  letter  to  Bentley)  for  any  one 
*  who  has  in  philosophical  matters  a  competent  faculty 
of  thinking '  for  a  moment  to  admit  the  possibility  of 
such  action."  J 

The  most  conclusive  evidence,  however,  of  the  re- 
pugnance between  the  assumption  of  distant  attraction 
and  the  elementary  concepts  of  mechanical  action  is 
found  in  the  incessant  renewal,  by  distinguished  men 
since  Newton's  day,  of  the  attempts  to  account  for  the 
phenomena  of  gravitation  on  the  principles  of  fluid 
pressure  or  solid  impact.*  These  attempts  have  recently 

*  "  Nonnisi  materia  ponderabili  interpositft  attractio  agere  potest." 
Geschichte  der  Erde,  Appendix,  p.  512. 

f  Ueber  die  Greuzen  des  Naturerkennens,  etc.,  p.  11. 
\  The  Unseen  Universe,  third  ed.  (1875),  p.  100. 

*  Some  of  these  attempts  are  very  ably  discussed  in  a  recent  memoir 
by  William  B.  Taylor  :  "  Kinetic  Theories  of  Gravitation,"  Smithsonian 
Report,  1876.     This  interesting  essay,  though  quite  exhaustive  in  the 
enumeration  of  the  theories  of  English  and  French  origin,  might  be  sup- 
plemented by  a  collection  of  references  to  German  articles  and  books  on 
the  same  subject.     See,  e.  g.,  Schramm,  "  Die  allgemeine  Bewegung  und 
Materie,"  Wien,  1872;  Aurel  Anderssohn,  "Die  Mechanik  der  Gravita- 


INERTIA  OF  ELEMENTS  OF  MASS.  £9 

become  reinvested  with  an  extraordinary  interest  in 
consequence  of  the  results  of  certain  experiments  of 
Professor  Guthrie,  who  found  that  light  bodies  sus- 
pended near  a  vibrating  disk  were  drawn  toward  it  "  as 
by  an  invisible  chord  " — a  phenomenon  which,  as  Sir 
William  Thomson  has  pointed  out,  is  explained  by  the 
fact  that  in  a  moving  fluid  the  pressure  is  least  where 
the  average  energy  of  motion  is  greatest.* 

In  the  eyes  of  modern  physics  all  modes  of  action 
which  appear  to  be  propagated  radially  from  a  center 
are  progressive  oscillations  in  elastic  media.  It  is  natu- 
ral, therefore,  to  look  for  the  physical  cause  of  gravita- 
tion in  the  same  direction.  Numerous  theories  have 
been  advanced  in  which  gravitation  is  referred  to  the 
wave-motion  of  an  elastic  interstellar  and  interatomic 
fluid  similar  to,  or  identical  with,  the  luminiferous 
aether.  The  most  noteworthy  of  these  theories  is  that 

tion,"  Breslau,  1874  (containing  a  photograph  of  the  results  of  an  ex- 
periment in  which  the  effe*cts  of  gravitation  are  simulated  by  a  ball  float- 
ing in  water  agitated  by  a  series  of  radial  impulses) ;  "  Zur  Loesung  des 
Problems  ueber  Sitz  und  Wesen  der  Anziehung" — 47  Versammlung 
deutscher  Natureforscher  und  Aerzte  zu  Breslau,  1874 ;  Hugo  Fritsch, 
"  Theorie  der  Newton'schen  Gravitation  und  des  Mariotte'schen  Gesetzes," 
Koenigsberg,  1874 ;  Ph.  Spiller,  "  Die  Urkraft  des  Weltalls,"  Berlin, 
1876,  etc.  It  is  somewhat  strange  that  Mr.  Taylor  ahould  have  omitted 
all  reference  to  Huygens's  elaborate  "  Dissertatio  de  causa  gravitatis  " 
(Hugenii,  Opp.  Reliqua,  vol.  i,  p.  95  seq.,  Amstelod.,  1728),  as  well  as  to 
the  equally  elaborate  theory  of  P.  A.  Secchi,  to  which  allusion  has  al- 
ready been  made  in  the  fourth  chapter.  In  our  own  country  Professor 
Pliny  Earle  Chase  has  made  large  contributions  to  this  class  of  literature. 
*  Guthrie's  experiments  had  been  anticipated,  without  his  knowledge, 
by  Guyot,  Schellbach,  and  others,  as  appears  from  a  communication 
of  Guthrie  himself  to  the  Philosophical  Magazine  (fourth  series,  vol. 
xli,  p.  405  seq.).  Experiments  similar  to  those  of  Aurel  Anderssohn 
were  made  long  ago  by  Hooke  and  Huygens,  both  of  whom  showed  that 
bodies  floating  on  water  agitated  by  waves  were  drawn  toward  the  center 
of  agitation.  Cf .  Hugenii,  "  Diss.  de  causa  gravitatis,"  Opp.  Reliqua,  i, 
p.  99  seq. 


60  CONCEPTS  OF  MODERN  PHYSICS. 

of  Professor  Challis,  who  assumes  that  all  space  is  filled 
with  a  vibrating  aether  which  "  is  a  continuous  elastic 
medium  perfectly  fluid  and  pressing  proportionally  to 
its  density."  Challis,  though  very  desirous  of  avoiding 
the  cumulation  of  hypothetical  media,  and  endeavoring 
to  construe  gravitative  action  as  an  incidental  or  resid- 
ual effect  of  the  luminar  and  thermal  vibrations  (resort- 
ing, for  this  purpose,  to  investigations  analogous  to 
those  of  Daniel  Bernoulli,  who  attempted,  more  than  a 
century  ago,  to  show  that  the  relative  motions  of  bodies 
composing  a  material  system  are  compounds  of  simple, 
regular,  and  permanent  oscillations  of  different  kinds) 
is  constrained  at  last  to  suggest  that  there  may  be 
an  aether  of  a  higher  order  "  having  the  same  relation 
to  the  first  as  that  has  to  air,  and  so  on  ad  libitum" 
and  that  "  the  form  of  gravity  is  due  to  the  attractive 
action  of  a  molecule  of  a  higher  order  as  to  magnitude 
than  the  molecule  of  molecular  attraction."  I  shall 
have  occasion,  in  a  subsequent  chapter,  to  discuss  the 
scientific  value  of  theories  of  this  sort,  in  which  facts 
are  explained  by  an  indefinite  number  of  arbitrary  as- 
sumptions multiplied  in  proportion  to  the  emergencies 
created  by  the  theories  themselves ;  for  the  present  it 
is  sufficient  to  observe  that  all  hydro-dynamical  theories 
of  gravitation  are  obnoxious  to  the  fatal  criticism  of 
Arago :  "  If  attraction  is  the  result  of  the  impulsion  of 
a  fluid,  its  action  must  employ  a  finite  time  in  traversing 
the  immense  spaces  which  separate  the  celestial  bod- 
ies," 'f  whereas  there  is  now  no  longer  any  reason  to 
doubt  that  the  action  of  gravity  is  instantaneous.  If  it 
were  otherwise — if  gravity,  like  light  or  electricity,  were 
propagated  with  a  measurable  velocity — there  would 
necessarily  be  a  composition  of  this  velocity  with  the 

*  Astronomie  populaire,  vol.  iv,  p.  119. 


INERTIA  OF  ELEMENTS   OF  MASS.  QY 

angular  orbital  velocities  of  the  planets  resulting  in 
their  acceleration  ;  the  apparent  line  of  attraction  would 
be  directed  to  a  point  in  advance  of  the  real  place  of 
the  sun,  just  as  the  sun's  apparent  position  is  displaced 
in  the  direction  of  the  earth's  orbital  motion  by  the 
aberration  of  light.  Such  an  effect,  if  it  had  any  exist- 
ence, would  have  been  detected  long  ago.  There  was 
a  time  when  the  action  of  gravity  was  supposed  to  be 
progressive.  Daniel  Bernoulli  attributed  the  non-coin- 
cidence of  the  tides  with  the  passage  of  the  moon 
through  the  meridian  to  the  comparative  slowness  of 
its  propagation;  and,  at  a  later  period,  Laplace  con- 
ceived for  a  moment  that  the  gradual  acceleration  of 
the  moon's  mean  motion  (first  ascertained  by  Halley  by 
a  comparison  of  modern  lunar  eclipses  with  those  re- 
corded by  Ptolemy  and  the  Arabian  astronomers)  might 
find  an  explanation  in  the  transmission  of  the  impulse 
of  gravity  with  a  velocity  exceeding  that  of  light  not 
less  than  eight  million  times.  But  the  retardation  of 
the  tides  is  now  known  to  be  a  consequence  of  the  iner- 
tia of  the  water  and  of  the  obstacles  which  it  encoun- 
ters in  its  flow;  and  the  acceleration  of  the  moon's 
motion  was  soon  shown,  by  Laplace  himself,  to  be 
caused,  in  great  part  at  least,  by  the  secular  diminution 
of  the  eccentricity  of  the  orbit  of  the  earth.  For  this 
reason  Laplace  did  not  hesitate  to  declare  that,  if  the 
action  of  gravity  was  propagated  in  time,  its  velocity 
must  be  at  least  fifty  million  times  greater  than  that  of 
light.  It  is  true  that  the  cause  assigned  by  him  for 
the  phenomenon  in  question  has  since  been  found  to  be 
inadequate  to  its  production.  From  a  revision  of  the 
calculations  of  the  French  astronomer  by  Mr.  Adams, 
some  years  ago,  it  appeared  that  the  diminution  of  the 
eccentricity  of  the  earth's  orbit  could  at  best  account 


62  CONCEPTS  OF  MODERN  PHYSICS. 

for  a  lunar  acceleration  of  six  seconds  in  one  century, 
instead  of  ten  seconds,  the  amount  of  acceleration  as- 
sumed by  Laplace,  and  furthermore,  that  the  accelera- 
tion amounted  in  fact  to  nearly  twelve  seconds.  A  part 
of  the  phenomenon,  therefore,  had  to  be  traced  to  other 
causes ;  and  this  has  been  successfully  effected  by  show- 
ing its  dependence  upon  the  tidal  retardation  of  the 
diurnal  motion  of  the  earth  which  occasions  an  appar- 
ent acceleration  of  the  mean  motion  of  the  moon. 

There  is  thus  an  entire  failure  of  analogy,  in  this 
respect,  between  the  action  of  gravity  and  the  other 
known  modes  of  physical  action  that  are  referred  to 
sethereal  undulations,  such  as  light,  radiant  heat  and 
electricity,  all  of  which  are  propagated  with  a  finite 
velocity.  There  are,  moreover,  as  Mr.  Taylor  has  ob- 
served, other  features  of  gravitation  which  give  rise  to 
the  presumption  that  it  is  of  a  nature  essentially  differ- 
ent from  that  of  other  forms  of  radial  action.  The 
action  of  gravity  is  wholly  unsusceptible  of  interfer- 
ence by  intervening  obstacles,  or,  as  Jevons  expresses 
it,*  "  all  bodies  are,  as  it  were,  absolutely  transparent 
to  it ; "  its  direction  is  in  right  lines  between  the  cen- 
ters of  the  attracting  masses,  and  is  not  subject  to  re- 
flection, refraction  or  composition;  unlike  the  forces 
of  cohesion,  capillarity,  chemical  affinity  and  electric  or 
magnetic  attraction,  it  is  incapable  of  exhaustion,  or 
rather  saturation,  every  body  attracting  every  other 
body  in  proportion  to  its  mass ;  it  is  wholly  indepen- 
dent of  the  nature,  volume,  or  structure  of  the  bodies 
between  which  it  occurs,  and  its  energy  is  unchange- 
able, incessant  and  inexhaustible. 

On  the  whole,  it  may  be  safely  said  that  the  undu- 
lations of  a  supposed  cosmical  aether  can  not  be  made 

*  Principles  of  Science,  vol.  ii,  p.  144. 


INERTIA  OF  ELEMENTS  OF  MASS.  $$ 

available  as  a  basis  for  a  physical  theory  of  gravitation, 
and  that,  if  such  a  theory  is  to  be  framed,  resort  must 
be  had  to  the  analogies  of  the  kinetic  theory  recently 
introduced  into  the  science  of  thermo-dynamics.  This 
is  very  frankly  admitted  by  leading  physicists  of  the 
day.  "  All  attempts  yet  made,"  say  Stewart  and  Tait,* 
"  to  connect  gravitation  with  the  luminiferous  aether, 
or  the  medium  required  to  explain  electric  and  mag- 
netic distance-action,  have  completely,  failed,  so  that 
we  are  apparently  driven  to  the  impact  theory  as  the 
only  possible  one."  The  only  impact  theory  seriously 
discussed  by  modern  physicists  and  astronomers  is  that 
of  Le  Sage,f  which,  stated  in  a  few  words,  is  this : 
Space  is  constantly  traversed  in  all  directions  by 
streams  of  infinitely  small  bodies  moving  with  an  al- 
most infinite  velocity  and  coming  from  unknown  re- 
gions of  the  universe.  These  bodies  are  termed 
"  ultramundane  corpuscules."  By  reason  of  their  mi- 
nuteness they  rarely,  if  ever,  collide,  and  the  greater 
part  of  them  find  ready  passage  through  ordinary 
sensible  bodies,  so  that  all  parts  of  these  bodies — those 
in  the  interior  as  well  as  those  on  the  surface — are 
equally  liable  to  be  struck  by  the  corpusciiles,  the  force 
of  the  impact  being  thus  proportional,  not  to,  the  sur- 
faces, but  to  the  masses  of  the  bodies.  A  single  body 
or  particle  would  be  equally  battered  by  these  corpus- 

*  The  Unseen  Universe,  §  140. 

f  Arago  suggests  (Astr.  pop.,  iv,  p.  118)  that  the  theory  of  Le  Sage 
is  simply  a  reproduction,  in  an  improved  form,  of  the  systematic  ideas  of 
Fatio  de  Duillers  (the  insane  and  meddlesome  partisan  of  Newton  in  his 
controversy  with  Leibnitz  respecting  the  priority  in  the  invention  of  the 
differential  calculus)  and  Varignon,  which  had  been  communicated  to 
Le  Sage  before  their  publication.  But  this  is  probably  an  error ;  Vari- 
gnon's  speculations,  at  least,  were  similar  to  those  of  Newton  in  the  21st 
Query  of  his  "  Opticks." 


64;  CONCEPTS  OF  MODERN  PHYSICS. 

cules  on  all  sides ;  but  any  two  bodies  act  as  ^mutual 
screens,  so  that  each  receives  a  less  number  of  impacts 
on  the  side  facing  the  other.  They  are  consequently 
driven  toward  each  other.  The  motion  of  the  corpus- 
cules  being  rectilinear  in  all  directions,  the  diminution 
of  pressure  thus  resulting  is  inversely  as  the  squares  of 
the  distances  between  the  bodies  affected. 

With  all  deference  due  to  the  authority  of  the  scien- 
tific men  by  whom  this  theory  has  been  countenanced, 
it  must  be  said  that  the  extravagance  of  its  assumptions 
at  once  characterizes  it  as  a  survival  of  the  fancies  of  an 
age  in  which  the  functions  of  a  scientific  theory  were 
imperfectly  .understood.  Its  intellectual  consanguinity 
with  the  old  vortices  and  harmonic  circulations  is  un- 
mistakable. It  utterly  ignores  the  necessity  of  account- 
ing for  the  origin  of  the  enormous  energy  constantly 
expended  by  the  supposed  streams  of  ultramundane 
corpuscules ;  both  the  agency  postulated  and  the  mode  of 
its  action  are  unknown  to  experience ;  and  it  is  doubt- 
ful whether  its  assumptions,  if  they  could  be  granted, 
would  serve  as  an  explanation  of  all  or  any  of  the  feat- 
ures of  gravitation  in  the  presence  of  which,  as  we  have 
seen,  every  hydro-dynamic  theory  is  doomed  to  failure. 
The  futility  of  Le  Sage's  theory,  however,  is  most  strik- 
ingly exhibited  by  Clerk  Maxwell,*  who  tests  it  by  the 
principle  of  the  conservation  of  energy.  If  the  ultra- 
mundane corpuscules  impinging  upon  sensible  bodies 
are  perfectly  elastic  and  rebound  with  the  same  velocity 
with  which  they  approach,  they  will  "  carry  their  energy 
with  them  into  the  ultramundane  regions,"  and  in  that 
event  "  the  corpuscules  rebounding  from  the  body  in 
any  given  direction  will  be  both  in  number  and  velocity 
exactly  equivalent  to  those  which  are  prevented  from 

*  Encyclopaedia  Britannica,  s.  v.  "  Atom." 


INERTIA  OF  ELEMENTS   OF   MASS.  (ft 

proceeding  in  that  direction  by  being  deflected  by  that 
body,  whatever  the  shape  of  the  body  and  however 
many  bodies  are  in  the  field."  In  this  case,  therefore, 
there  is  no  gravitative  action.  If,  on  the  other  hand, 
the  corpuscules  are  inelastic,  or  imperfectly  elastic — 
inasmuch  as  the  action  of  gravity  is  supposed  to  be  due 
to  the  comparatively  small  difference  between  the  im- 
pacts on  opposite  sides  of  the  body — the  energy  of  those 
impacts,  at  least,  which  balance  each  other,  must  be 
(partially  or  wholly,  according  to  the  degree  of  corpus- 
cular elasticity)  converted  into  heat,  and  "  the  amount 
of  heat  so  generated  would  in  a  few  seconds  raise  the 
body,  and  in  like  manner  the  whole  material  universe, 
to  a  white  heat."  * 

Once  more,  then,  science  is  in  irreconcilable  conflict 
with  one  of  the  fundamental  postulates  of  the  mechani- 
cal theory.  Action  at  a  distance,  the  impossibility  of 
which  the  theory  is  constrained  to  assert,  proves  to  be 
an  ultimate  fact  inexplicable  on  the  principles  of  impact 
and  pressure  of  bodies  in  immediate  contact.  And  this 
fact  is  the  foundation  of  the  most  magnificent  theoreti- 
cal structure  which  science  has  ever  erected — a  founda- 
tion deepening  with  every  new  reach  of  our  telescopic 
vision,  and  broadening  with  every  further  stretch  of 
mathematical  analysis. 

*  Mr.  S.  Tolver  Preston  has  recently  (Phil.  Mag.,  September  and 
November,  1877,  and  February  and  May,  1878)  proposed  a  modification 
of  Le  Sage's  theory,  in  which  he  seeks  to  dispense  with  the  ultramun- 
dane teature  of  the  corpuscules,  and  to  account  for  gravitation  upon  the 
postulate  of  the  kinetic  theory  of  gases  alone.  His  theory  is  founded  on 
the  assumptions  that  "  the  range  of  gravity  is  limited,"  and  that  "  the 
stars  move  in  straight  lines  and  not  in  orbits."  In  view  of  these  assump- 
tions, and  of  my  discussion  of  the  kinetic  theory  of  gases  in  a  separate 
chapter,  I  do  not  deem  it  necessary  to  devote  any  space  to  it  here. 


CHAPTEE  YI. 

THE     PROPOSITION     THAT  ALL     POTENTIAL     ENERGY     IS    IN 

REALITY    KINETIC. EVOLUTION    OF     THE    DOCTRINE    OF 

THE    CONSERVATION   OF    ENERGY. 

ACCORDING  to  the  mechanical  theory,  motion,  like 
mass,  is  indestructible  and  unchangeable ;  it  can  not 
vanish  and  reappear.  Any  change  in  its  rate  results 
from  its  distribution  among  a  greater  or  less  number  of 
units  of  mass.  And,  motion  and  mass  being  mutually 
inconvertible,  nothing  but  motion  can  be  the  cause  of 
motion.  There  is,  therefore,  no  potential  energy ;  all 
energy  is  in  reality  kinetic. 

The  close  logical  connection  of  this  proposition  with 
that  discussed  in  the  last  chapter  is  obvious,  and  has 
not  escaped  the  notice  of  leading  physicists.  Stewart 
and  Tait,  after  giving  an  account  of  Le  Sage's  hypothe- 
sis, which,  in  their  opinion,  contains  the  rudiments,  at 
least,  of  the  only  tenable  physical  theory  of  gravita- 
tion, proceed  to  say :  u  If  Le  Sage's  theory,  or  any- 
thing of  a  similar  nature,  be  at  all  a  representation  of 
the  mechanism  of  gravitation,  a  fatal  blow  is  dealt 
to  the  motion  of  the  tranquil  form  of  power  we  have 
called  potential  energy.  Not  that  there  will  cease  to 
be  a  profound  difference  in  kind  between  it  and  ordi- 
nary kinetic  energy,  ~but  that  BOTH  will  ~be  henceforth 
to  ~be  regarded  as  kinetic"  *  This  declaration  has  re- 

*  The  Unseen  Universe,  §  142. 


CONSERVATION  OF  ENERGY.  Qf 

cently  been  repeated  by  Professor  Tait  in  his  lecture 
on  Force.* 

The  proposition  here  insisted  upon  is  irrecusable  by 
any  consistent  advocate  of  the  mechanical  theory.  But, 
again,  modern  science  peremptorily  refuses  its  assent. 
It  asserts  that  all,  or  nearly  all,  physical  changes  in  the 
universe  are  mutual  conversions  of  kinetic  and  poten- 
tial energies — that  energy  is  incessantly  stored  as  virtual 
power  and  restored  as  actual  motion.  When  the  bob 
of  an  ordinary  pendulum  descends  from  its  highest  to 
its  lowest  point,  its  potential  energy  diminishes  in  pro- 
portion to  the  increase  of  its  actual  motion;  when  it 
rises  again,  its  energy  of  motion  disappears  at  the  same 
rate  up  to  its  arrival  at  its  highest  point  opposite  the 
first,  where  it  is  for  an  instant  motionless,  all  its  energy 
being  due  to  its  position.  And  this  conversion  and  re- 
conversion of  the  two  forms  of  energy  are  typical  alike 
of  the  supposed  oscillations  of  the  ultimate  atoms  or 
molecules  and  of  the  orbital  swing  of  the  large  bodies 
composing  a  planetary  system.  A  planet  moving  in  an 
excentric  orbit  gains  energy  of  motion  as  it  approaches 
the  sun  and  loses  it  again  in  the  same  proportion  as  it 
recedes  from  it.  The  same  mutual  transformation  is 
exhibited  in  another  wide  domain  of  physical  phe- 
nomena :  action  due  to  chemical  affinity.  A  lump  of 
coal  lies  buried  in  the  earth  for  a  million  years ;  during 
all  this  time  there  is  no  appreciable  change  in  its  posi- 
tion as  referred  to  surrounding  objects,  or  in  the  relative 
positions  of  its  parts — it  is  without  external  or  internal 
motion  (except  that  which  it  shares  with  the  planet  of 
which  it  is  a  part)  ;  now  we  bring  it  to  the  surface,  into 
the  atmosphere  containing  oxygen  and  into  contact  with 

*  On  some  Recent  Advances  in  Physical  Science,  second  ed.,  pp.  262, 
263 


(J8  CONCEPTS  OF  MODERN  PHYSICS. 

a  flame ;  its  latent  power  at  once  becomes  sensible — it 
burns,  giving  rise  to  vigorous  action  which  manifests 
itself  as  light  and  heat.v  The  tendency  of  modern  sci- 
ence is  to  trace  all  physical  change  to  a  few  primary 
forms  of  potential  energy,  chief  among  which  are  grav- 
ity and  chemical  affinity.  In  the  opinion  of  modern 
physicists,  the  only  plausible  theory,  thus  far  advanced, 
of  the  origin  of  stellar  and  planetary  systems  is  that 
known  as  the  nebular  hypothesis;  and,  whether  we 
adopt  its  familiar  Kant-Laplacean  form,  or  one  of  its 
more  recent  modifications,  in  either  case  all  the  molar, 
if  not  the  molecular,  forces  of  the  universe  are  ulti- 
mately derived  from  the  attraction  due  to  the  mere 
position  of  the  original  particles  supposed  to  be  uni- 
formly diffused  in  space.  And  all  changes  in  the 
comparatively  minute  organic  or  inorganic  forms  are 
referred,  proximately  at  least,  in  physiology  as  well 
as  in  physics,  to  the  affinities  of  the  chemical  ele- 
ments. ' 

In  truth,  modern  science  teaches  that  diversity  and 
change  in  the  phenomena  of  nature  are  possible  only  on 
condition  that  energy  of  motion  is  capable  of  being 
stored  as  energy  of  position.  The  relatively  perma- 
nent concretion  of  material  forms,  chemical  action  and 
reaction,  crystallization,  the  evolution  of  vegetal  and 
animal  organisms — all  depend  upon  the  "  locking  up  " 
of  kinetic  action  in  the  form  of  latent  energy.  To 
make  this  clear,  and  to  show  that  the  effort  to  abolish 
the  distinction  between  kinetic  and  potential  energy  is 
without  avail,  it  will  be  useful  briefly  to  review  the 
history  of  the  doctrine  of  the  conservation  of  energy. 

In  a  general  sense,  this  doctrine  is  coeval  with  the 
dawn  of  human  intelligence.  It  is  nothing  more  than 
an  application  of  the  simple  principle  that  nothing  can 


CONSERVATION   OF  ENERGY.  (J9 

come  from  or  to  nothing.*  But  the  history  of  its  de- 
velopment and  adhibition  in  physical  science  begins  with 
its  emphatic  statement  in  the  ';  Principia  Philosophiae  " 
of  the  inventor  of  the  system  of  cosmical  vortices.f 

*  It  may  be  truly  asserted  that  human  intelligence  begins  and  ends 
with  the  principle  above  stated.  When  all  the  phenomenal  changes  in 
the  universe  shall  have  been  reduced  to  the  one  principle  of  the  conser- 
vation of  energy,  the  time  will  have  come  for  celebrating  the  final  con- 
summation of  physical  science  in  a  new  epic  "  de  rerum  naturd  ;  "  and  in 
its  first  chapter  will  again  be  written  the  words  of  Lucretius : 

".  .  .  res  .  .  .  non posse creari 
De  niMlo,  neque  item  genitas  in  nil  revocari." 

It  is  not  a  little  curious  to  note  the  unaminity  and  emphasis  with  which 
the  early  Greek  philosophers  gave  utterance  to  the  declaration  that  noth- 
ing could  absolutely  originate  or  perish — the  rudimentary  form  of  the 
law  of  causality.  Diogenes  of  Apollonia  declared :  "  ovfev  e/c  rov  \u\ 
uvros  ylveffbcu"  (Diog.  Laert.,  ix,  57);  Parmenides:  "&s  ayeverov  cbv 
KOI  w(&\€^pov  fonv"  (Karsten,  Rel,  v,  58);  Empedocles :  "e/c  rov 
yap  uri  %6vros  a.p.-t)xav<>v  €'°"T'  yevfff&cu  "  (Karsten,  v,  48);  Dcmocritos : 
"  /tiqSev  r'  £K  rov  ^  ovros  yivecr&ou  KaL  e*s  rb  ^  *ov  Q&etpes&cu "  (Diog. 
Lacrt.,  ix,  44).  The  first  application  of  this  principle  to  motion  was 
made  by  Epicurus  (Diog.  Laert.,  lib.  x ;  Lucret.  "  De  rer.  nat.,"  w.  294 
-307),  who  sought  to  demonstrate  the  conservation  of  both  mass  and 
motion  by  the  argument  that  there  is  no  place  beyond  the  universe  to 
which  matter  or  motion  could  be  communicated  or  from  which  it  could 
be  derived — an  argument  which  was  reproduced  by  Leibnitz  (Opp. 
Math.,  vol.  vi,  p.  440— cf.  Berthold,  "Notizen,"  etc.,  in  Pogg.  Ann., 
vol..  clvii,  p.  342),  and  which  is  in  effect  a  shrewd  anticipation  of  the 
modern  concept  of  a  "  conservative  system."  An  elaborate  exposition 
of  the  Epicurean  doctrine  is  given  by  Gassendi  ("  Ad  librum  decimum 
Diogenis  Laertii  Notae,"  opp.,  ed.  Lugd.,  vol.  iii,  p.  241  seq.).  It  is  not 
improbable  that  this  exposition  had  its  influence  on  the  meditations  of 
Descartes,  notwithstanding  the  wide  divergence  between  his  philosophi- 
cal tendencies  and  those  of  Gassendi. 

f  Descartes  has  been  called  the  father  of  modern  philosophy ;  with 
equal  propriety  he  might  also  be  designated  as  the  father  of  modern  phys- 
ical science.  His  title  to  the  honors  of  paternity  in  philosophy,  no  less 
than  in  physics,  must  find  other  muniments  than  the  discovery,  or  even 
exact  statement,  of  permanently  valuable  truths.  Few  of  his  philosophical 
tenets  endure,  at  least  in  the  form  in  which  he  held  them,  and  some  of 
the  truths  which  he  rejected  are  now  counted  among  our  most  indispen- 
4 


70  CONCEPTS  OF  MODERN  PHYSICS. 

Descartes  announced  the  doctrine  of  the  conserva- 
tion of  motion  in  terms  perfectly  explicit.  He  declared 
that  God  was  the  spring-head  of  motion,  and  always 
conserved  in  the  world  the  same  quantity  of  motion.* 

sable  possessions.  As  a  physicist  he  broached  a  number  of  theories  that 
have  proved  to  he  wholly  unfounded,  and  he  ignored  or  misconceived  al- 
most all  the  laws  of  mechanical  action  whose  discovery  constituted  the 
distinction  of  his  older  contemporary,  Galilei.  In  philosophy  he  was  the 
immediate  progenitor  of  Spinoza,  whose  system,  though  in  effect  an  in- 
voluntary reductio  ad  absurdum  of  all  ontological  speculation,  has  served, 
by  reason  of  the  specious  elegance  of  its  pseudo-mathematical  paralo- 
gisms, to  retard  the  discovery  of  true  principles  of  philosophical  inquiry 
to  an  incalculable  extent.  In  physics  his  vagaries  obscured  the  field  of 
investigation  to  such  a  degree  that  the  shadows  have  not  wholly  vanished 
to  this  day.  Though  professing  to  emancipate  himself  from  the  meta- 
physical traditions  of  the  period  which  was  then  near  its  close,  he  was 
thoroughly  imbued  with  their  spirit.  But  precisely  for  this  reason  his 
writings  influenced  the  thought  of  the  seventeenth  century  more  exten- 
sively than  the  researches  of  those  who  resorted  to  the  scientific  methods 
of  experiment  and  observation — methods  that  were  wholly  at  variance 
with  the  mental  habits  of  the  age.  He  was  essentially  a  metaphysician, 
an  ontologist  of  the  mediaeval  type ;  but  he  discussed  nearly  all  the  prob 
lems  whose  solution  was  the  task  devolving  upon  the  physicists  and 
mathematicians  of  the  two  centuries  that  have  elapsed  since  his  day. 
Thus  his  speculations,  though  on  the  whole  nugatory  hi  themselves,  be- 
came the  ferment  which  induced  the  process  of  gradual  clarification  in 
the  rapidly  thickening  mixture  of  scientific  material.  This  ferment  was 
not  the  less  important  because  it  was  almost  wholly  lost  in  the  progress 
of  its  action. 

In  saying  all  this  I  have  no  disposition  to  detract  from  the  general 
admiration  due  to  the  vigor  and  acuteness  of  his  intellect ;  nor  do  I  for- 
get that  he  is  the  founder  of  analytical  geometry.  And  it  is  not  neces- 
sary, I  trust,  to  add  that,  while  I  give  candid  expression  to  my  estimate 
of  the  value  of  Spinoza's  philosophical  system,  I  am  not  a  stranger  to  the 
emotion  which  will  always  be  felt  when  the  touching  figure  of  the  lonely 
thinker  rises  into  view,  and  that  I  am  not  insensible  to  the  charm  of  the 
simple  beauty  of  a  life  which,  more  perfectly,  perhaps,  than  any  other, 
exemplifies  the  Tusculan  definition :  vivere  est  cogitare. 

*  "  Generalem  (motus  causam)  quod  attinet,  manifestum  mihi  videtur 
illam  non  aliam  esse  quam  Deum  ipsum  qui  materiam  simul  cum  motu 
et  quiete  in  principle  creavit,  jamque  per  solum  suum  concursum  ordina- 


CONSERVATION  OF  ENERGY.  ft 

If  he  had  not  been  precluded  (by  his  assumption  that 
the  only  primary  properties  of  matter  were  extension 
and  mobility)  from  admitting  the  atomic  constitution 
of  matter,  he  would,  no  doubt,  have  asserted  the  con- 
servation of  motion  in  the  sense  which  is  generally  at- 
tributed to  the  principle  of  the  conservation  of  energy 
in  our  day  by  persons  without  scientific  training :  that 
the  atoms  of  which  the  material  world  is  composed  are 
perpetually  in  a  state  of  uniform  translatory  or  oscilla- 
tary  motion,  changing  only  in  direction,  or,  if  they  move 
with  different  velocities,  that  the  sum  of  these  velocities 
is  constant.  In  view  of  his  general  physical  theory, 
Descartes  was  constrained  to  resort,  not  to  the  atom — 
the  supposed  primordial  unit  of  mass,  the  existence  of 
which  he  denied — but  to  mass  generally  ;  and  the  con- 
servation of  motion  in  his  system  assumed  the  form  of 
a  conservation  of  the  quantity 'of  motion  in  the  sense  of 
the  sum  of  the  products  of  all  masses  into  their  respec- 
tive velocities.*  It  is  worthy  of  note  that  the  term 
"  quantity  of  motion  "  as  expressive  of  the  product  of  a 
mass  into  its  velocity  (i.  e.,  momentum)  was  adopted  by 
Newton,  and  has  maintained  itself  in  physics  to  the 
present  day. 

It  is  manifest  that  the  conservation  of  motion,  as  an 

rium,  tantundem  motus  et  quietis  in  ea  tota,  quantum  tune  posuit,  con- 
servat."  Princ.  Phil.,  ii,  §  36.  The  doctrine  is  stated,  substantially  in 
the  same  terms,  in  various  other  parts  of  the  same  work,  e.  g.,  ii,  §  42 ; 
iii,  §  46. 

*  The  vagueness  of  Descartes's  mechanical  notions  is  strikingly  ex- 
hibited in  his  efforts  to  reconcile  this  with  his  third  law  of  motion,  ac- 
cording to  which  a  body  loses  no  motion  in  a  collision  with  a  "  stronger  " 
one — "  ubi  corpus  quod  movetur  alteri  occurrit,  si  minorem  habeat  vim  ad 
pergendum  secundum  lineam  rectam,  quam  hoc  alterum  ad  ei  resistendum, 
et  motum  suum  retinendo  solam  motus  determinationem  amittit ;  si  vero 
habeat  majorem,  tune  alterum  corpus  secum  movet  ac  quantum  ei  dat  de 
suo  motu,  tantundem  perdit."  Princ.  Phil.,  ii,  §  40. 


Y2  CONCEPTS  OF  MODERN  PHYSICS. 

absolute  quantity  in  the  popular  sense  (in  which  it  is,  in 
fact,  a  conservation  of  velocities),  would  be  possible 
only  in  a  world  without  differences  of  density  or  struct- 
ure. If  motion  were  conserved  in  this  sense,  there 
could  be  neither  phenomenal  diversity,  nor  phenomenal 
change.  To  the  universe  as  we  know  it,  with  its  inces- 
sant transformations,  the  assumed  principle  of  the  con- 
servation of  motion  can  have  no  application.  This  was 
seen,  dimly  at  least,  by  Leibnitz,  who  denied  that  there 
was  any  conservation  of  motion  in  the  Cartesian  sense. 
His  denial  found  its  most  pointed  expression  in  an 
essay  entitled  "  Short  demonstration  of  the  memorable 
error  of  Descartes  and  others  in  regard  to  a  law  of  nat- 
ure, according  to  which,  as  they  claim,  God  always  con- 
serves the  same  quantity  of  motion,  which  they  also 
abuse  in  mechanics."  *  To  the  Cartesian  doctrine  of 
the  conservation  of  the  quantity  of  motion  he  opposed 
the  principle  of  the  conservation  of  vis  viva — of  the 
product  of  mass  into  the  square  of  its  velocity. 

Here  was  the  origin  of  the  famous  controversy  be- 
tween the  Leibnitians  and  Cartesians,  respecting  the  true 
measure  of  the  forces  in  the  universe,  which  was  par- 
ticipated in  by  so  many  mathematicians  and  philoso- 
phers, and  to  wrhich,  as  is  well  known,  a  late  and  inap- 
posite contribution  was  made  by  Kant.  This  contro- 
versy has  long  since  been  finally  settled ;  but  it  is  so 
important  for  my  ulterior  purpose  to  clear  up  the 
prevalent  misconceptions  of  the  true  import  of  the 
principle  of  the  conservation  of  energy,  that  I  devote 
a  moment's  consideration  to  its  merits. 

*  "  Brevis  dcmonstratio  erroris  memorabilis  Oartesii  ct  aliorum  circa 
legem  naturae,  secundum  guam  volunt  a  Deo  eandem  semper  quantilatem 
motus  conservari,  qud  et  in  re  mechanicd  abutuntur"  Acta  Erud.,  Lips., 
1686  (Leibn.,  opp.  math.,  vol.  vi,  p.  117). 


CONSERVATION  OF   ENERGY.  73 

Force  in  its  ordinary  sense  (as  the  cause  of  motion, 
or  rather,  as  the  aggregate  of  all  its  conditions)  finds  its 
measure  simply  in  the  velocity  of  a  unit  of  mass.  Thus 
force  and  mass  are  measured  by  each  other.  Two 
forces  are  the  same  when  they  generate  the  same  ve- 
locity (or,  more  generally,  the  same  acceleration)  in  the 
same  mass ;  and  two  masses  are  the  same  when  they  are 
equally  accelerated  by  the  same  force.  When  the  mo- 
tion of  a  unit  of  mass  is  distributed  among  several 
units,  the  motion  of  each  unit  becomes  less  in  propor- 
tion to  the  number  of  units  among  which  the  distribu- 
tion is  made.  The  velocity  (or  acceleration)  of  a  body 
is  therefore  directly  as  the  force,  and  inversely  as  the 
mass.  And,  in  the  case  of  constant  forces  producing 
uniform  accelerations,  the  velocities  are  obviously  pro- 
portional to  the  times  of  action. 

We  have,  therefore, 

Force      „.. 
velocity  =  — X  Time  of  Action,  or, 

Mass 

Mass  X   Velocity  =  Force  X  Time  of  Action ;  .  .  .  (1) 

i.  e.,  the  force  exerted  during  any  given  time  is  equal 
to  the  product  of  the  mass  into  the  velocity.  On  the 
other  hand,  the  space  or  distance  through  which  a  body 
moves  under  the  action  of  a  constant  force  is,  like  veloc- 
ity, directly  as  the  force  and  inversely  as  the  mass ;  but, 
unlike  velocity,  it  is  proportional,  not  to  the  time  simply, 
but  to  half  the  square  of  the  time  of  action.  Hence, 

Force 
Space  or  distance  of  Action  =  —      -  X  i  (Time  of  Action)2, 

Mass 
or  (inasmuch  as,  according  to  the  first  equation, 

m.          ,.  A  Mass  X  Velocity  \ 

Time  of  Action  =  -  — ^  J 

Force          / 

£  Mass  X  Velocity 2  =  Force  X  Distance  of  Action  „ . .  (2). 


» 
74  CONCEPTS  OF  MODERN  PHYSICS. 

The  first  term  of  this  last  equation — the  product 
of  the  mass  into  half  the  square  of  the  velocity — is 
the  Leibnitian  vis  viva,  and  is  now  termed  kinetic 
energy.* 

It  is  apparent  that  the  first  (Cartesian)  formula  in- 
dicates the  measure  of  a  given  force  during  a  given  time 
of  action,  while  the  second  (Leibnitian)  formula  con- 
tains the  measure  of  the  force  acting  through  a  given 
distance.  There  is  no  inconsistency  between  the  two ; 
on  the  contrary,  the  one  is  a  corollary  from  the  other. 
And  yet  the  controversy  is  of  interest  in  view  of  the 
Cartesian  claim  (which  survives  as  an  indelible  fancy 
in  many  minds)  that  force,  in  the  sense  of  the  rate  of 
the  generation  or  transference  of  quantity  of  motion,  is 
conserved,  and  that  the  momenta  during  any  two  equal 
intervals  of  time  are  the  same.  In  the  light  of  modern 
science  nothing  is  more  demonstrably  untrue  than  the 
doctrine  of  the  conservation  of  motion  as  it  was  held 
by  Descartes.  Nevertheless,  there  is  a  sense  in  which 
the  quantity  of  motion — or  what  is  now  usually  called 
momentum — is  constant  in  the  mutual  actions  of  bodies 
composing  a  material  system.  Momentum  being  the 
product  of  mass  into  the  velocity,  and  velocity  being 
necessarily  in  a  definite  direction,  it  follows,  as  New- 
ton himself  has  shown,  from  his  third  law  (according 
to  which  action  and  reaction  are  equal  and  opposite — 
all  force,  so  called,  being  but  one  aspect  of  the  mutu- 
al equal  and  opposite  action  of  two  bodies — )  that  the 
momentum  of  any  system  of  bodies,  i.  e.,  the  sum  of 
their  quantities  of  motion,  in  whatever  direction  these 
quantities  be  measured,  is  never  changed  by  their  mut- 

*  Leibnitz  and  his  contemporaries  designated  the  whole  product  of  the 
mass  into  the  square  of  the  velocity  as  the  vis  viva  ;  but  this  is  correct 
only  when  the  measure  of  forces  is  stated  in  the  form  of  a  proportion. 


CONSERVATION  OF  ENERGY.  f£ 

ual  action.  Whatever  momentum  is  acquired  by  any 
part  of  the  system  is  lost  by  another  part  in  the  same 
direction.  From  this  follows  the  important  dynamical 
principle  (announced  in  Newton's  fourth  corollary  from 
his  laws  of  motion)  that  the  center  of  inertia  of  a  sys- 
tem of  bodies  is  never  affected  by  their  mutual  ac- 
tion. 

To  interpret  the  Cartesian  proposition  in  its  appli- 
cation to  the  universe  as  a  single  conservative  system, 
so  as  to  make  it  conformable  to  fact,  it  would  be  neces- 
sary to  take  some  one  fixed  direction  and  project  upon 
it  all  the  motions  of  its  constituent  bodies  or  particles 
— in  other  words,  to  take  their  effective  components  as 
represented  by  the  cosines  of  the  angles  between  their 
several  directions  and  the  standard  direction  to  which 
they  are  referred.  This  being  done,  the  sum  of  the 
momenta,  i.  e.,  of  the  products  of  all  the  masses  into 
their  velocities  in  the  direction  indicated,  would  be 
constant;  it  being  understood  that,  if  motion  in  one 
direction  is  taken  as  positive,  motion  in  the  opposite 
direction  (and  hence  also  the  momentum  whereof  it  is 
a  factor)  is  negative.* 

Although  the  merit  of  having  formulated  the  prin- 
ciple of  the  conservation  of  vis  viva  belongs  to  Leib- 
nitz, the  first  clear  statement  of  the  relation  of  this 
principle  to  that  of  the  conservation  of  momentum  is 

*  It  is  sometimes  said  that  quantities  of  motion  partially  or  wholly 
neutralize  or  destroy  each  other,  as  in  the  case  of  the  central  collision  of 
two  inelastic  bodies  moving  with  equal  velocities  in  diametrically  oppo- 
site directions,  where  the  bodies,  after  impact,  are  at  rest,  the  resultant 
momentum  being  =  0.  But  the  momenta  of  the  two  bodies  being  equal 
and  opposite,  and  their  sum,  therefore,  being  that  of  two  equal  quantities, 
one  of  which  is  positive  and  the  other  negative,  this  sum  was  also  =  0 
before  collision,  so  that  the  case  stated  is  no  exception  to  the  rule  that 
the  momenta  of  colliding  bodies  are  unaltered  by  their  mutual  impact. 


Yg  CONCEPTS  OF  MODERN  PHYSICS. 

due  to  Huygens,  and  is  in  these  words :  "  The  quantity 
of  motion  possessed  by  two  bodies  may  be  augmented 
or  diminished  by  their  encounter;  but  there  remains 
always  the  same  quantity  on  the  same  side,  if  we  sub- 
tract the  quantity  of  opposite  motion.  .  .  .  The  sum 
of  the  products  of  every  hard  body  multiplied  by  the 
square  of  its  velocity  is  always  the  same  before  and 
after  the  encounter."* 

The  progress  made  up  to  this  point,  in  the  rectifi- 
cation of  the  Cartesian  doctrine,  consisted  in  the  denial 
of  the  conservation  of  motion  in  the  sense  of  mere 
velocity  or  of  the  quantity  of  motion  and  the  rate  of 
its  change  irrespective  of  its  direction,  and  in  the  asser- 
tion of  the  conservation  of  energy  of  motion — a  quan- 
tity proportional  to  the  product  of  mass  into  the  square 
of  its  velocity.  Such  was  the  state  of  the  doctrine  in 
Newton's  time. 

The  Leibnitian  principle  might,  even  at  this  time, 
(all  the  premisses  being  given  in  Newton's  laws  of  mo- 
tion, and  especially  in  his  interpretation  of  the  third 
law)  have  been  generalized  so  as  to  embrace,  or  to  im- 

*  "  La  quantity  du  mouvement  qu'ont  deux  corps  se  peut  augmenter 
ou  diminuer  par  leur  rencontre ;  mais  il  y  reste  toujours  la  meme  quan- 
tite  vers  le  meme  cote,  en  soustrayant  la  quantite  du  mouvement  con- 
traire  ....  La  somme  des  produits  f  aits  de  la  grandeur  de  chaque  corps 
dur  multiplie  par  le  quarre"  de  sa  vitesse,  est  toujours  la  meme  devant  et 
apres  la  rencontre."  Cf.  Aikin,  "  On  the  History  of  Force,"  Phil.  Mag., 
4th  series,  vol.  xxviii,  p.  472.  Professor  Bonn  (ib.,  p.  313)  claims  the 
honor  of  priority  in  giving  a  clear  exposition  of  the  principle  of  the 
conservation  of  vis  viva  for  John  Bernoulli ;  but  upon  perusal  of  the 
passages  quoted  by  him  it  will  be  seen  that  Bernoulli's  conception  rested 
upon  the  metaphysical  assumption  of  the  substantiality  of  motion  and 
the  equality  of  cause  and  effect.  Indeed,  John  Bernoulli  had  adopted 
the  principle  in  the  form  and  upon  the  considerations  presented  by  Leib- 
nitz, who,  like  Descartes,  was  a  metaphysician  rather  than  a  physicist, 
while  Huygens,  a  true  man  of  science,  arrived  at  his  propositions  by  a 
series  of  generalizations  of  special  cases. 


CONSERVATION  OF  ENERGY.  -77 

ply,  not  only  the  conservation  of  vis  viva,  but  also  the 
principle  of  virtual  velocities,  the  conservation  of  mo- 
mentum (including  angular  momentum)  and  the  mod- 
ern principle  of  the  conservation  of  energy.  The  for- 
mula would  have  been  this:  Neither  the  momentum, 
nor  the  energy,  of  a  system  of  bodies  is  ever  changed 
by  their  mutual  actions.  It  is  manifest  that  this  is 
nothing  more  than  an  extension  of  the  principle  of  in- 
ertia according  to  which  a  body,  whether  it  be  regarded 
as  simple  or  as  composed  of  parts,  can  not  move  itself, 
i.  e.,  can  not  produce  any  change  in  its  own  state  of 
rest  or  of  uniform  motion  as  a  whole. 

Modern  science  has  framed  a  number  of  concepts 
which  serve  to  facilitate  the  apprehension  of  the  laws 
regulating  changes  in  the  condition  of  material  aggre- 
gates. Treating  every  sensible  body  as  a  system  of 
units  of  mass,  it  defines  "  work "  as  a  change  in  the 
configuration  of  such  a  system  in  opposition  to  the 
forces  resisting  it,  and  "  energy  "  as  the  capacity  to  do 
work.  Whenever  such  a  system  is  considered  as  being 
under  the  exclusive  control  of  the  mutual  forces  of  its 
constituent  units,  i.  e.,  when  it  is  neither  acted  upon  by 
other  systems,  nor  acts  upon  them,  it  is  called  a  "  con- 
servative system."  In  fact  there  is  no  limited  material 
system  which  is  not  involved  in  mutual  actions  with 
systems  or  bodies  without,  and  for  this  reason  a  "  con- 
servative system "  is  more  appropriately  defined  as  a 
group  of  bodies  which,  in  passing  through  any  cycle  of 
changes  of  configuration,  does  the  same  quantity  of  ex- 
ternal work  which  is  done  upon  it,  so  that  the  energy 
derived  from  bodies  without  is  compensated  for  by  an 
equal  amount  of  energy  communicated  to  external 
bodies.  If,  now,  we  express  the  principle  of  the  con- 
servation of  vis  viva  in  terms  of  these  concepts,  it  as- 


78  CONCEPTS  OF  MODERN  PHYSICS. 

sumes  the  following  form :  In  any  series  of  changes  in 
the  configuration  of  a  conservative  system,  its  actual 
energy  (energy  of  motion,  or  vis  viva — now  termed  ki- 
netic energy)  is  the  same  whenever  the  configuration  is 
the  same,  i.  e.,  whenever  its  constituent  units  are  in  the 
same  relative  positions,  through  whatever  orbits  and  with 
whatever  velocities  they  may  have  moved  in  the  passage 
from  one  configuration  to  the  other.  The  import  of 
this  proposition  will  be  best  realized  by  considering  the 
simple  case  of  the  oscillations  of  a  pendulum  which, 
ever  since  the  days  of  Galileo,  has  served  as  a  paradigm 
for  the  illustration  of  dynamical  laws.  The  bob  of  the 
pendulum  changes  velocity  at  every  point;  but  the 
velocities  at  points  equidistant  from  its  point  of  maxi- 
mum velocity  are  equal.*  A  still  simpler  case  is  that  of 
a  body  projected  perpendicularly  upward  and  return- 
ing to  the  point  from  which  it  was  projected  j  in  its 
ascent  it  is  retarded,  and  in  its  descent  accelerated  (leav- 
ing out  of  account  the  resistance  of  the  air),  by  the 
constant  action  of  gravity ;  but  at  the  same  points  the 
velocities  of  ascent  and  descent  are  the  same.  A  simi- 
lar (at  bottom  the  same)  instance  is  afforded  by  celestial 
bodies,  moving  in  elliptical  orbits,  which — again  abstract- 
ing from  causes  that  interfere  with  the  strict  periodicity 
of  their  motions — have  the  same  energies  of  motion  at 
the  same,  or  symmetrically  corresponding,  points  of 
their  orbits.  The  instances  here  adduced  are  all  cases 
of  varying  (uniformly  accelerated  or  retarded)  mo- 
tion; when  the  motion  is  uniform,  the  law  of  con- 
servation is  simply  the  well-known  principle  of  virtual 
velocities. 

Obviously  the  next  question  in  order  is  :  What  is  the 

*  This  is,  of  course,  strictly  true  only  of  an  ideal  pendulum,  swing- 
ing in  vacua  and  without  friction. 


CONSERVATION  OF  ENERGY.  f  9 

law  of  energy  without  regard  to  the  completion  of  the 
cycle  of  configurative  changes — in  the  interval,  during 
the  passage  of  the  system  from  any  assumed  initial  con- 
figuration to  a  different  one,  and  during  its  return  from 
this  to  the  initial  configuration?  The  answer  to  this 
question,  which  has  taken  definite  form  in  very  recent 
times,  constitutes  the  true  and  exhaustive  statement  of 
the  doctrine  of  the  conservation  of  energy.  x/Et  is  this :  In  / 
any  series  of  changes  in  the  configuration  of  a  conserv^-sl 
ative  system,  the  sum  of  its  kinetic  and  potential  ener-  ' 
gies  (i.  e.,  the  actual  energy  of  the  system  at  a  given  in- 
stant added  to  the  work  done  in  passing  from  the  initial 
configuration  to  the  configuration  at  that  instant)  is  con- 
stant— the  work  done  being  stored  as  power  to  re- 
produce the  initial  configuration  and  thus  to  restore  the 
actual  energy  lost.  Literally,  this  statement  of  the 
principle  applies  only  to  cases  where  work  is  done 
against  the  forces  of  the  system,  as,  for  instance,  when 
a  body  is  projected  upward  against  gravity — when, 
therefore,  kinetic  energy  is  stored  as  potential  energy. 
Whenever,  conversely,  kinetic  energy  is  restored  and 
potential  energy  lost,  as  in  the  case  of  a  falling  body, 
the  statement  must  be  so  modified  as  to  assert  the  con- 
stancy of  the  sum  obtained  by  adding  the  kinetic 
energy  due  to  a  given  configuration  to  the  work  to  le 
done  in  reproducing  the  initial  configuration  where  the 
potential  energy  is  at  its  maximum.  In  such  cases  the 
mathematical  expression  for  the  potential  energy  in 
terms  of  work  is  negative.  In  its  application  to  the 
energy  of  the  universe  (which  is  necessarily  conserv- 
ative, there  being  no  bodies  without  it)  the  law  of  con- 
servation is  this:  The  kinetic  energy  of  the  universe 
plus  the  work  to  be  done  by  the  mutual  forces  of  its 
constituent  elements  by  removing  them  to  the  limit  of 


80  CONCEPTS  OF  MODERN  PHYSICS. 

exhaustion  of  the  action  of  these  forces,  i.  e.,  to  infinite 
distances  from  each  other,  is  at  all  times  constant.* 

The  conformity  of  the  principle  of  the  conservation 
of  energy  to  the  facts  of  experience  is  sufficiently  ap- 
parent whenever  we  deal  with  visible  or  otherwise  per- 
ceptible changes  in  the  position  or  configuration  of  a 
body  or  system  of  bodies,  such  as  the  action  of  gravity, 
the  strain  of  an  elastic  body,  etc.  In  these  cases  we 
readily  see  that  energy  is  alternately  stored  as  energy 
of  position  and  restored  as  energy  of  motion.  But 
there  is  a  class  of  cases  in  which  there  is  loss  of  energy 
of  motion  without  manifest  change  of  position.  "When 
two  equal  inelastic  bodies,  moving  with  equal  veloci- 
ties in  opposite  directions,  collide  centrally,  there  is, 
apparently  at  least,  a  total  destruction  of  motion,  and 
there  is  no  gain  of  position,  for  the  bodies  remain  at 
rest  at  the  point  of  collision.  A  similar  loss  of  actual 
energy  is  observed  whenever  work  is  done  against  fric- 
tion. What  becomes  of  the  energy  of  motion  which 
seems  to  disappear  in  cases  of  this  kind?  To  this 
question  Newton  clearly  had  no  definite  answer.  He 
expressly  asserted  that  "  motion  may  be  got  or  lost," 
and  that,  "  the  vis  inertice  being  a  passive  principle, 
.  .  .  some  other  principle  was  necessary  for  putting 
bodies  into  motion,  and,  now  they  are  in  motion,  some 
other  principle  is  necessary  for  conserving  the  motion. 
.  •.  .  By  reason  of  the  tenacity  of  fluids,  and  attrition 

*  It  is  to  be  observed  that  I  am  here  stating  the  doctrine  of  the  con- 
servation of  energy  in  its  application  to  the  universe  as  it  is  generally 
held  among  physicists.  The  discussion  of  the  question  respecting  the 
admissibility  of  applying  logical  concepts  and  mathematical  formulae 
based  upon  the  conditions  of  finite  existence  to  the  Infinite,  of  dealing 
with  the  boundless  world  as  with  a  definite  mechanical  system,  and  with 
its  energy  as  with  a  constant  quantity,  must  be  reserved  for  a  later  stage 
in  the  progress  of  our  inquiry. 


CONSERVATION  OF  ENERGY.  ^1 

of  their  parts,  and  the  weakness  of  elasticity  in  solids, 
motion  is  much  more  apt  to  be  lost  than  got,  and  is 
always  upon  the  decay."  *  But  it  is  an  error  to  main- 
tain, with  Stewart  and  Tait,f  that  the  answer  was  un- 
known in  Newton's  time.  The  answer  of  modern  sci- 
ence, which  is  that  the  apparent  loss  of  molar  motion 
results  from  its  real  conversion  into  molecular  motion, 
was  anticipated  by  Leibnitz,  as  is  shown  in  the  follow- 
ing remarkable  passage  found  in  his  fifth  letter  to 
Clarke :  "  I  had  maintained  that  the  active  forces  are 
conserved  in  the  world.  It  is  objected  that  two  soft 
or  inelastic  bodies,  when  they  collide,  lose  part  of  their 
force.  I  answer,  they  do  not.  It  is  true  that  the 
4  wholes  '  lose  it  in  reference  to  their  total  movement ; 
but  it  is  received  by  the  particles,  they  being  agitated 
inwardly  by  the  force  of  the  collision.  Thus  the  loss 
ensues  only  in  appearance.  The  forces  are  not  de- 
stroyed, but  dissipated  among  the  minute  parts.  This 
is  not  losing  them,  but  it  is  doing  what  those  do  who 
turn  money  into  small  change."  ^  '  The  truth  thus  an- 

*  "  Opticks,"  4th  ed.,  p,  373. 

f  The  Unseen  Universe,  §  100. 

\  "  J'avais  soutenu  que  les  Forces  actives  sc  conservent  dans  le  monde. 
On  m'objecte,  que  deux  corps  moux,  ou  non-elastiques,  concourant  entre 
eux,  perdent  de.  leur  force.  Je  reponds  que  non.  II  est  vrai  que  les 
Touts  la  perdent  par  rapport  &  leur  mouvement  total ;  mais  les  parties 
la  re9oivent,  etant  agitees  interieurement  par  la  force  du  concours. 
Ainsi  ce  defaut  n'arrive  qu'en  appareuce.  Les  forces  ne  sont  detruites, 
mais  dissipees  parmi  les  parties  menues.  Ce  n'est  pas  les  perdre,  mais 
c'est  faire  comme  font  ceux  qui  changent  la  grosse  monnaie  en  petite." 
Opp.  phil.,  ed.  Erdmann,  p.  775.  It  is  strange  that  this  passage  should 
have  remained  unnoticed  for  many  years  even  after  the  adoption  of  the 
modern  theory  of  the  conservation  and  transformation  of  energy  and  of 
the  correlation  of  forces.  I  found  it  many  years  ago  ;  Du  Bois-Rey- 
mond  has  recently  called  attention  to  it  in  a  lecture,  "  Leibnizische  Ge- 
danken  in  der  neueren  Naturwissenschaft."  There  is  another  passage  of 
the  same  import  in  Leibnitz's  Mathematical  Works  (ed.  Gerhardt),  vol. 


82  CONCEPTS  OF  MODERN   PHYSICS. 

nounced  was  a  "  bedridden  truth  "  (to  use  an  expression 
of  Coleridge)  for  a  long  time  ;  in  spite  of  the  vigorous, 
and  even  violent,  disputes  about  forces  and  their  meas- 
ure, and  in  the  midst  of  the  rapid  accumulation  of 
physical  facts  and  theories,  it  remained  barren  for  more 
than  a  century.  This  seemingly  anomalous  fact  is  ex- 
plained by  the  circumstance  that,  up  to  the  middle  of 
the  present  century,  heat,  electricity,  magnetism,  etc., 
were  supposed  to  be  material  substances  whose  intercon- 
vertibility  with  mechanical  motion  or  energy  appeared 
to  be  utterly  inconceivable.  It  was  only  after  the  es- 
tablishment of  the  dynamic  theories  of  the  "  imponder- 
ables "  that  the  doctrine  of  the  conservation  and  trans- 
formation of  energy  became  fertile,  and  led  to  a  fun- 
damental reconstitution  of  the  entire  body  of  physics.* 
The  correlation  and  mutual  conversion  of  the  various 
forms  of  energy  have  been  so  extensively  illustrated  in 
the  scientific  writings  of  the  day,  that  it  is  unnecessary 
to  dwell  upon  them  here.  The  purpose  of  my  hurried 
glance  at  the  history  of  the  doctrine  of  the  conserva- 
tion of  energy,  or  rather,  of  the  evolution  of  the  sci- 
entific concepts  embraced  in  it,  was  simply  to  show 
that  this  history  is  in  effect  that  of  a  progressive  aban- 
donment of  the  mechanical  proposition  placed  at  the 
head  of  the  present  chapter,  which  is  substantially  iden- 
tical with  Descartes's  theory  of  the  conservation  of  mo- 

ii,  p.  230.  Dr.  Berthold  has  shown  (Pogg.  Ann.,  vol.  clvii,  p.  350) 
that  the  "allotropy  of  force"  was  announced,  more  than  a  century  ago, 
in  terms  of  curious  precision,  by  Diderot  in  his  "  Pensees  sur  1' interpre- 
tation de  la  nature,"  Londres,  1754,  §  45. 

*  I  ain  aware,  of  course,  of  the  anticipations  of  the  modern  theory 
of  heat  by  Bacon,  Locke,  Rumford,  Sir  Humphry  Davy,  etc. ;  but  their 
announcement,  however  clear,  that  heat  is  but  a  "  mode  of  motion,"  re- 
ceived as  little  attention  from  contemporary  physicists  as  the  Leibnitian 
doctrine  above  referred  to. 


CONSERVATION  OF  ENERGY.  Qg 

tion — a  circumstance  whose  significance  I  hope  to  point 
out  hereafter. 

We  have  now  discussed  the  four  cardinal  proposi- 
tions of  the  atomo-mechanical  theory,  and  have  found 
(without  entering  upon  the  domain  of  the  organic  sci- 
ences) that  they  are  severally  denied  by  the  sciences  of 
chemistry,  physics,  and  astronomy.  Before  we  proceed 
to  investigate  the  causes  and  consequences  of  this  result 
and  to  consider  the  relation  of  the  mechanical  theory 
to  the  laws  of  thought  and  the  history  of  its  evolution, 
it  is  important  to  supplement  this  discussion  by  an  in- 
quiry into  the  nature,  validity,  and  scientific  value  of 
the  hypothesis  of  the  atomic  constitution  of  matter. 


CHAPTEE  VII. 

THE   THEORY   OF   THE   ATOMIC    CONSTITUTION   OF   MATTER. 

THE  doctrine  that  an  exhaustive  analysis  of  matter 
into  its  real  elements,  if  it  could  be  practically  effected, 
would  yield  an  aggregate  of  indivisible  and  indestructi- 
ble particles,  is  one  of  the  earliest  products  of  human 
speculation,  and  has  held  its  ground  more  persistently 
than  any  other  tenet  of  science  or  philosophy.  It  is 
true  that  the  atomic  theory,  since  its  first  promulgation 
by  the  old  Greek  philosophers,  and  its  elaborate  state- 
ment by  Lucretius,  has  been  modified  and  refined. 
There  is  probably  no  one,  at  this  day,  who  invests  the 
atoms  with  hooks  and  loops,  or  accounts  for  the  bitter 
taste  of  wormwood  by  the  raggedness,  and  for  the 
sweetness  of  honey  and  milk  by  the  smooth  roundness 
of  the  constituent  atoms.*  But  the  atoms  of  modern 
science  are  still  of  determinate  weight,  if  not  of  defi- 
nite, uniform  and  constant  figure',  and  stand  for  some- 
thing more  than  abstract  units  even  in  the  view  of  those 
who,  like  Boscovich,  Faraday,  Ampere,  or  Fechner,  pro- 
fess to  regard  them  as  mere  centers  of  force.  And 
there  is  no  difficulty  in  stating  the  atomic  doctrine  in 
terms  applicable  alike  to  all  the  acceptations  in  which 
it  is  now  held  by  scientific  men.  Whatever  diversity 
of  opinion  may  prevail  as  to  the  form,  size,  etc.,  of  the 
atoms,  all  who  advance  the  atomic  hypothesis,  in  any  of 

*  Lucretius,  DC  Rerum  Nat.,  ii,  398  seq. 


THE   ATOMIC   THEORY.  '85 

its  varieties,  as  a  physical  theory,  agree  in  three  propo- 
sitions, which  may  be  stated  as  follows : 

1.  Atoms  are  absolutely  simple,  unchangeable,  inde- 
structible ;  they  are  physically,  if  not  mathematically, 
indivisible. 

2.  Matter  consists  of  discrete  parts,  the  constituent 
atoms  being  separated  by  void  interstitial  spaces.     In 
contrast  to  the  continuity  of  space  stands  the  disconti- 
nuity of  matter.    TJie  expansion  of  a  body  is  simply  an 
increase,  its  contraction  a  lessening,  of  the  spatial  inter- 
vals between  the  atoms. 

3.  The  atoms  composing  the  different  chemical  ele- 
ments are  of  determinate  specific  weights,  corresponding 
to  their  equivalents  of  combination* 

Confessedly  the  atomic  theory  is  but  an  hypothesis. 
This  in  itself  is  not  decisive  against  its  value ;  all  phys- 
ical theories  properly  so  called  are  hypotheses  whose 
eventual  recognition  as  truths  depends  upon  their  con- 
sistency with  themselves,  upon  their  agreement  with 
the  canons  of  logic,  upon  their  congruence  with  the 
facts  which  they  serve  to  connect  and  explain,  upon 
their  conformity  with  the  ascertained  order  of  Nature, 
upon  the  extent  to  which  they  approve  themselves  as 
trustworthy  anticipations  or  previsions  of  facts  verified 
by  subsequent  observation  or  experiment,  and  finally 
upon  their  simplicity,  or  rather  their  reducing  power. 
The  merits  of  the  atomic  theory,  too,  are  to  be  deter- 
mined by  seeing  whether  or  not  it  satisfactorily  and 
simply  accounts  for  the  phenomena  as  the  explanation 
of  which  it  is  propounded,  and  whether  or  not  it  is  in 

*  To  avoid  confusion,  I  purposely  ignore,  for  the  moment,  the  distinc- 
tion between  molecules  as  the  ultimate  products  of  the  physical  division 
of  matter,  and  atoms  as  the  ultimate  products  of  its  chemical  decomposi- 
tion, preferring  to  use  the  word  atoms  in  the  sense  of  the  least  particles 
into  which  bodies  are  divisible  by  any  means. 


gg  CONCEPTS  OF  MODERN  PHYSICS. 

accord  with  itself  and  with  the  known  laws  of  Reason 
and  of  Nature. 

For  what  facts,  then,  is  the  atomic  hypothesis  meant 
to  account,  and  to  what  degree  is  the  account  it  offers 
satisfactory  ? 

xx  It  is  claimed  that  the  first  of  the  three  propositions 
above  enumerated  (the  proposition  which  asserts  the 
persistent  integrity  of  atoms,  or  their  un  changeability 
both  in  weight  and  volume)  accounts  for  the  indestruc- 
tibility and  impenetrability  of  matter ;  that  the  second 
of  these  propositions  (relating  to  the  discontinuity  of 
matter)  is  an  indispensable  postulate  for  the  explana- 
tion of  certain  physical  phenomena,  such  as  the  disper- 
sion and  polarization  of  light ;  and  that  the  third  propo- 
sition (according  to  which  the  atoms  composing  the 
chemical  elements  are  of  determinate  specific  gravities) 
is  the  necessary  general  expression  of  the  laws  of  defi- 
nite constitution,  equivalent  proportion,  and  multiple 
combination,  in  chemistry. '' 

In  discussing  these  claims,  it  is  important,  first,  to 
verify  the  facts  and  to  reduce  the  statements  of  these 
facts  to  exact  expression,  and  then  to  see  how  far  they 
are  fused  by  the  theory. 

1.  The  indestructibility  of  matter  is  an  unquestion- 
able truth.  But  in  what  sense,  and  upon  what  grounds, 
is  this  indestructibility  predicated  of  matter  ?  The  unan- 
imous answer  of  the  atomists  is :  Experience  teaches 
that  all  the  changes  to  which  matter  is  subject  are  but 
variations  of  form,  and  that  amid  these  variations  there 
is  an  unvarying  constant — the  mass  or  quantity  of  mat- 
ter. The  constancy  of  the  mass  is  attested  by  the  bal- 
ance, which  shows  that  neither  fusion  nor  sublimation, 
neither  generation  nor  corruption,  can  add  to  or  detract 
from  the  weight  of  a  body  subjected  to  experiment. 


THE   ATOMIC   THEORY.  £7 

When  a  pound  of  carbon  is  burned,  the  balance  demon- 
strates the  continuing  existence  of  this  pound  in  the 
carbonic  acid,  which  is  the  product  of  combustion,  and 
from  which  the  original  weight  of  carbon  may  be  re- 
covered. The  quantity  of  matter  is  measured  by  its 
weight,  and  this  weight  is  unchangeable. 

Such  is  the  fact,  familiar  to  every  one,  and  its  in- 
terpretation equally  familiar.  To  test  the  correctness 
of  this  interpretation,  we  may  be  permitted  slightly  to 
vary  the  method  of  verifying  it.  Instead  of  burning 
the  pound  of  carbon,  let  us  simply  carry  it  to  the  sum- 
mit of  a  mountain,  or  remove  it  to  a  lower  latitude  ;  is 
its  weight  still  the  same  ?  Relatively  it  is  ;  it  will  still 
balance  the  original  counterpoise.  But  the  absolute 
weight  is  no  longer  the  same.  This  appears  at  once,  if 
we  give  to  the  balance  another  form,  taking  a  pendu- 
lum instead  of  a  pair  of  scales.  The  pendulum  on  the 
mountain  or  near  the  equator  swings  more  slowly  than 
at  the  foot  of  the  mountain  or  near  the  pole,  for  the 
reason  that  it  has  become  specifically  lighter  by  being 
farther  removed  from  the  center  of  the  earth's  attrac- 
tion, in  conformity  to  the  law  that  the  attractions  of 
bodies  vary  inversely  as  the  squares  of  their  distances. 

It  is  thus  evident  that  the  constancy,  upon  the  ob- 
servation of  which  the  assertion  of  the  indestructibility 
of  matter  is  based,  is  simply  the  constancy  of  a  relation, 
and  that  the  ordinary  statement  of  the  fact  is  crude  and 
inadequate.  Indeed,  while  it  is  true  that  the  weight  of 
a  body  is  a  measure  of  its  mass,  this  is  but  a  single  case 
of  the  more  general  fact  that  the  masses  of  bodies  are 
inversely  as  the  velocities  imparted  to  them  by  the 
action  of  the  same  force,  or,  more  generally  still,  in- 
versely as  the  accelerations  produced  in  them  by  the 
same  force.  In  the  case  of  gravity,  the  forces  of  attrac- 


gg  CONCEPTS  OF  MODERN  PHYSICS. 

tion  are  directly  proportional  to  the  masses,  so  that  the 
action  of  these  forces  (weight)  is  the  simplest  measure 
of  the  relation  between  any  two  masses  as  such ;  but,  in 
any  inquiry  relating  to  the  validity  of  the  atomic  the- 
ory, it  is  necessary  to  bear  in  mind  that  this  weight  is 
not  the  equivalent,  or  rather  presentation,  of  an  abso- 
lute substantive  entity  in  one  of  the  bodies  (the  body 
weighed),  but  the  mere  expression  of  a  relation  between 
two  bodies  mutually  attracting  each  other.  And  it  is 
further  necessary  to  remember  that  this  weight  may  be 
indefinitely  reduced,  without  any  diminution  in  the 
mass  of  the  body  weighed,  by  a  mere  change  of  its 
position  in  reference  to  the  body  between  which  and 
the  body  weighed  the  relation  subsists. 

Masses  find  their  true  and  only  measure  in  the  ac- 
tion of  forces,  and  the  persistence  of  the  effect  of  this 
action  is  the  simple  and  accurate  expression  of  the  fact 
which  is  ordinarily  described  as  the  indestructibility  of 
matter.  It  is  obvious  that  this  persistence  is  in  no  sense 
explained  or  accounted  for  by  the  atomic  hypothesis. 
It  may  be  that  such  persistence  is  an  attribute  of  the 
minute,  insensible  particles  which  are  supposed  to  con- 
stitute matter,  as  well  as  of  sensible  masses ;  but,  surely, 
the  hypothetical  recurrence  of  a  fact  in  the  atom  is  no 
explanation  of  the  actual  occurrei^ce  of  the  same  fact  in 
the  conglomerate  mass.  Whatever  mystery  is  involved 
in  the  phenomenon  is  as  great  in  the  case  of  the  atom 
as  in  that  of  a  solar  or  planetary  sphere.  Breaking  a 
magnet  into  fragments,  and  showing  that  each  fragment 
is  endowed  with  the  magnetic  polarity  of  the  integer 
magnet,  is  no  explanation  of  the  phenomenon  of  mag- 
netism. A  phenomenon  is  not  explained  by  being 
dwarfed.  A  fact  is  not  transformed  into  a  theory  by 
being  looked  at  through  an  inverted  telescope.  The 


THE   ATOMIC   THEORY.  $9 

hypothesis  of  ultimate  indestructible  atoms  is  not  a 
necessary  implication  of  the  persistence  of  weight,  and 
can  at  best  account  for  the  indestructibility  of  matter 
if  it  can  be  shown  that  there  is  an  absolute  limit  to  the 
compressibility  of  matter — in  other  words,  that  there  is 
an  absolutely  least  volume  for  every  determinate  mass. 
This  brings  us  to  the  consideration  of  that  general  prop- 
erty of  matter  which  probably,  in  the  minds  of  most 
men,  most  urgently  requires  the  assumption  of  atoms — 
its  impenetrability. 

"  Two  bodies  can  not  occupy  the  same  space  " — this 
is  the  ordinary  statement  of  the  fact  in  question.  Like 
the  indestructibility  of  matter,  it  is  claimed  to  be  a 
datum  of  experience.  "  That  all  bodies  are  impene- 
trable," says  Sir  Isaac.  Newton,  "  we  gather,  not  from 
reason,  but  from  sense."*  Let  us  see  in  what  sense 
and  to  what  extent  this  claim  is  legitimate. 

The  proposition,  according  to  which  a  space  occu- 
pied by  one  body  can  not  be  occupied  by  another,  im- 
plies the  assumption  that  space  is  an  absolute,  self- 
measuring,  objective  entity,  and  the  further  assumption 
that  there  is  a  least  space  which  a  given  body  will  abso- 
lutely fill  so  as  to  exclude  any  other  body.  A  verifica- 
tion of  this  proposition  by  experience,  therefore,  must 
amount  to  proof  that  there  is  an  absolute  limit  to  the 
compressibility  of  all  matter  whatsoever.  Now,  does 
experience  authorize  us  to  assign  such  a  limit  ?  As- 
suredly not.  It  is  true  that  in  the  case  of  solids  and 
liquids  there  are  practical  limits  beyond  which  compres- 
sion by  the  mechanical  means  at  our  command  is  im- 
possible ;  but  even  here  we  are  met  by  the  fact  that  the 
volumes  of  fluids,  which  effectually  resist  all  efforts  at 

*  "  Corpora  omnia  impenetrabilia  esse,  non  rationc,  sed  sensu  colligi- 
mus." — Phil.  Nat.  Princ.  Math.,  lib.  iii,  reg.  3. 


90  CONCEPTS  OF  MODERN  PHYSICS. 

further  reduction  by  external  pressure,  are  readily  re- 
duced by  mere  mixture.  Thus,  sulphuric  acid  and 
water  at  ordinary  temperatures  do  not  sensibly  yield  to 
pressure ;  but,  when  they  are  mixed,  the  resulting  vol- 
ume is  materially  less  than  the  aggregate  volumes  of 
the  liquids  mixed.  But,  waiving  this,  as  well  as  the 
phenomena  which  emerge  in  the  processes  of  solution 
and  chemical  action,  it  must  be  said  that  experience 
does  not  in  any  manner  vouch  for  the  impenetrability 
of  matter  in  all  its  stages  of  aggregation.  When  gases 
are  subjected  to  pressure,  the  result  is  simply  an  increase 
of  the  expansive  force  in  proportion  to  the  pressure  ex- 
erted, according  to  the  law  of  Boyle  or  Mariotte  (the 
modifications  of  and  apparent  exceptions  to  which,  as 
exhibited  in  the  experimental  results  obtained  by  Re- 
gnault  and  others,  need  not  here  be  stated,  because 
they  do  not  affect  the  argument).  A  definite  experi- 
mental limit  is  reached  in  the  case  of  those  gases  only 
in  which  the  pressure  produces  liquefaction  or  solidifi- 
cation. The  most  significant  phenomenon,  however, 
which  experience  contributes  to  the  testimony  on  this 
subject,  is  the  diffusion  of  gases.  Whenever  two  or 
more  gases  which  do  not  act  upon  each  other  chemical- 
ly are  introduced  into  a  given  space,  each  gas  diffuses 
itself  in  this  space  as  though  it  were  alone  present 
there ;  or,  as  Dalton,  the  reputed  father  of  the  modern 
atomic  theory,  expresses  it,  "  Gases  are  mutually  pas- 
sive, and  pass  into  each  other  as  into  vacua." 

Whatever  reality  may  correspond  to  the  notion  of 
the  impenetrability  of  matter,  this  impenetrability  is 
not,  in  the  sense  of  the  atomists,  a  datum  of  experience. 

Upon  the  whole,  it  would  seem  that  the  validity  of 
the  first  proposition  of  the  atomic  theory  is  not  sus- 
tained by  the  facts.  Even  if  the  assumed  unchange- 


THE  ATOMIC   THEORY.  ^1 

ability  of  the  supposed  ultimate  constituent  particles  of 
matter  presented  itself,  upon  its  own  showing,  as  more 
than  a  bare  reproduction  of  an  observed  fact  in  the 
form  of  an  hypothesis,  and  could  be  dignified  with  the 
name  of  a  generalization  or  of  a  theory,  it  would  still 
be  obnoxious  to  the  criticism  that  it  is  a  generalization 
from  facts  crudely  observed  and  imperfectly  appre- 
hended. 

In  this  connection  it  may  be  noted  that  the  atomic 
theory  has  become  next  to  valueless  as  an  explanation 
of  the  impenetrability  of  matter,  since  it  has  been 
pressed  into  the  service  of  the  undulatory  theories  of 
radiance,  and  assumed  the  form  in  which  it  is  now  held 
by  the  majority  of  physicists,  as  we  shall  presently  see. 
According  to  this  form  of  the  theory,  the  atoms  are 
either  mere  points,  wholly  without  extension,  or  their 
dimensions  are  infinitely  small  as  compared  with  the 
distances  between  them,  whatever  be  the  state  of  aggre- 
gation of  the  substances  into  which  they  enter.  In 
this  view  the  resistance  which  a  body,  i.  e.,  a  system  of 
atoms,  offers  to  the  intrusion  of  another  body  is  due, 
not  to  the  rigidity  or  unchangeability  of  volume  of  the 
individual  atoms,  but  to  the  relation  between  the  at- 
tractive and  repulsive  forces  with  which  they  are  sup- 
posed to  be  endowed.  There  are  physicists  holding 
this  view  who  are  of  opinion  that  the  atomic  constitu- 
tion of  matter  is  consistent  with  its  penetrability — 
among  them  M.  Cauchy,  who,  after  defining  atoms  as 
"material  points  without  extension,"  uses  this  lan- 
guage :  "  Thus,  this  property  of  matter  which  we  call 
impenetrability  is  explained,  when  we  consider  the 
atoms  as  material  points  exerting  on  each  other  attrac- 
tions and  repulsions  which  vary  with  the  distances  that 
separate  them.  .  .  .  From  this  it  follows  that,  if  it 


92  CONCEPTS  OF  MODERN  PHYSICS. 

pleased  the  author  of  Nature  simply  to  modify  the  laws 
according  to  which  the  atoms  attract  or  repel  each 
other,  we  might  instantly  see  the  hardest  bodies  pene- 
trating each  other,  the  smallest  particles  of  matter  oc- 
cupying immense  spaces,  or  the  largest  masses  reduc- 
ing themselves  to  the  smallest  volumes,  the  entire  uni- 
verse concentrating  itself,  as  it  were,  in  a  single  point."  * 

2.  The  second  fundamental  proposition  of  the  mod- 
ern atomic  theory  avouches  the  essential  discontinuity 
of  matter.  The  advocates  of  the  theory  affirm  that 
there  is  a  series  of  physical  phenomena  which  are  inex- 
plicable, unless  we  assume  that  the  constituent  particles 
of  matter  are  separated  by  void  interspaces.  The  most 
notable  among  these  phenomena  are  the  dispersion  and 
polarization  of  light.  The  grounds  upon  which  the 
assumption  of  a  discrete  molecular  structure  of  matter 
is  deemed  indispensable  for  the  explanation  of  these 
phenomena  may  be  stated  in  a  few  words. 

According  to  the  undulatory  theory,  the  dispersion 
of  light,  or  its  separation  into  spectral  colors,  by  means 
of  refraction,  is  a  consequence  of  the  unequal  retarda- 
tion experienced  by  the  different  waves,  which  produce 
the  different  colors,  in  their  transmission  through  the 
refracting  medium.  This  unequal  retardation  presup- 

"  Ainsi,  cette  propriete  de  la  matiere  que  nous  nommons  impene- 
trabilite  se  trouve  expliquee,  quand  on  considere  les  atomes  comme  des 
points  materials  qui  exercent  les  uns  sur  les  autres  des  attractions  ou 
repulsions  variables  avec  les  distances  qui  les  separent.  ...  II  requite 
encore  de  ce  qui  precede,  que  s'il  plaisait  &  1'auteur  de  la  nature  de  modi- 
fier seulement  les  lois  suivant  lesquelles  les  atomes  s'attirent  ou  se  re- 
poussent,  nous  pourrions  voir,  &  1'instant  meme,  les  corps  les  plus  durs  se 
penetrer  les  uns  les  autres,  les  plus  petites  parcelles  de  matiere  occuper 
des  espaces  demesures,  ou  les  masses  les  plus  considerables  se  reduire 
aux  plus  petits  volumes,  et  1'univers  entier  se  concentrer  pour  ainsi  dire 
en  un  seul  point."  Sept  Le9ons  de  Physique  GSnerale,  ed.  Moigno,  p. 
38  seq. 


THE  ATOMIC  THEORY.  "93 

poses  differences  in  the  velocities  with  which  the  various 
colored  rays  are  transmitted  through  any  medium  what- 
ever, and  a  dependence  of  these  velocities  upon  the 
lengths  of  the  waves.  But,  according  to  a  well-estab- 
lished mechanical  theorem,  the  velocities  with  which 
undulations  are  propagated  through  a  continuous  me- 
dium depend  solely  upon  the  elasticity  of  the  medium 
as  compared  with  its  inertia,  and  are  wholly  independ- 
ent of  the  length  and  form  of  the  waves.  The  correct- 
ness of  this  theorem  is  attested  by  experience  in  the 
case  of  sound.  Sounds  of  every  pitch  travel  with  the 
same  velocity.  If  it  were  otherwise,  music  heard  at  a 
distance  would  evidently  become  chaotic;  differences 
of  velocity  in  the  propagation  of  sound  would  entail  a 
distortion  of  the  rhythm,  and,  in  many  cases,  a  reversal 
of  the  order  of  succession.  Now,  differences  of  color 
are  analogous  to  differences  of  pitch  in  sound,  both 
reducing  themselves  to  differences  of  wave-length. 
The  lengths  of  the  waves  increase  as  we  descend  the 
scale  of  sounds  from  those  of  a  higher  to  those  of  a 
lower  pitch;  and  similarly,  the  length  of  a  luminar 
undulation  increases  as  we  descend  the  spectral  scale, 
from  violet  to  red.  It  follows,  then,  that  the  rays  of 
different  color,  like  the  sounds  of  different  pitch,  should 
be  propagated  with  equal  velocities,  and  be  equally  re- 
fracted ;  that,  therefore,  no  dispersion  of  light  should 
take  place. 

This  theoretical  impossibility  of  dispersion  has  al- 
ways been  recognized  as  one  of  the  most  formidable 
difficulties  of  the  undulatory  theory.  In  order  to  ob- 
viate it,  Cauchy,  at  the  suggestion  of  his  friend  Coriolis, 
entered  t  upon  a  series  of  analytical  investigations,  in 
which  he  succeeded  in  showing  that  the  velocities  with 
which  the  several  colored  rays  are  propagated  may  vary 
5 


94  CONCEPTS  OF  MODERN  PHYSICS. 

according  to  the  wave-lengths,  if  it  be  assumed  that  the 
sethereal  medium  of  propagation,  instead  of  being  con- 
tinuous, consists  of  particles  separated  by  sensible  dis- 
tances. 

By  means  of  a  similar  assumption,  Fresnel  has  sought 
to  remove  the  difficulties  presented  by  the  phenomena 
of  polarization.  In  ordinary  light,  the  different  undu- 
lations are  supposed  to  take  place  in  different  directions, 
all  transverse  to  the  course  or  line  of  propagation,  while 
in  polarized  light  the  vibrations,  though  still  transverse 
to  the  ray,  are  parallelized,  so  as  to  occur  in  the  same 
plane.  Soon  after  this  hypothesis  had  been  expanded 
into  an  elaborate  theory  of  polarization,  Poisson  ob- 
served that,  at  any  considerable  distance  from  the  source 
of  the  light,  all  transverse  vibrations  in  a  continuous 
elastic  medium  must  become  longitudinal.  As  in  the 
case  of  dispersion,  this  objection  was  met  by  the  hy- 
pothesis of  the  existence  of  "  finite  intervals  "  between 
the  sethereal  particles. 

These  are  the  considerations,  succinctly  stated,  which 
theoretical  physics  are  supposed  to  bring  to  the  support 
of  the  atomic  theory.  In  reference  to  the  cogency  of 
the  argument  founded  upon  them,  it  is  to  be  said,  gen- 
erally, that  evidence  of  the  discrete  molecular  arrange- 
ment of  matter  is  by  no  means  proof  of  the  alternation 
of  unchangeable  and  indivisible  atoms  with  absolute 
spatial  voids.  But  it  is  to  be  feared  that  the  argument 
in  question  is  not  only  formally,  but  also  materially, 
fallacious.  It  is  very  questionable  whether  the  assump- 
tion of  "  finite  intervals  "  between  the  particles  of  the 
luminiferous  aether  is  competent  to  relieve  the  undula- 
tory  theory  of  light  from  its  embarrassments.  This 
subject,  in  one  of  its  aspects,  has  been  thoroughly  dis- 
cussed by  E.  B.  Hunt,  in  an  article  on  the  dispersion  of 


X 


THE  ATOMIC  THEORY.  95 

light,*  and  the  suggestions  there  made  appear  to  me 
worthy  of  serious  attention.     They  are  briefly  these  : 

M.  Cauchy  brings  the  phenomena  of  dispersion  within 
the  dominion  of  the  undulatory  theory,  by  deducing  the 
differences  in  the  velocities  of  the  several  chromatic  rays 
from  the  differences  in  the  corresponding  wave-lengths 
by  means  of  the  hypothesis  of  definite  intervals  between 
the  particles  of  the  light-bearing  medium.  He  takes  it 
for  granted,  therefore,  that  these  chromatic  rays  are  prop- 
agated with  different  velocities.  But  is  this  the  fact  ? 
Astronomy  affords  the  means  to  answer  this  question. 

We  experience  the  sensation  of  white  light  when 
all  the  chromatic  rays  of  which  it  is  composed  strike  the 
eye  simultaneously.  The  light  proceeding  from  a  lu- 
minous body  will  appear  colorless,  even  if  the  compo- 
nent rays  move  with  unequal  velocities,  provided  all 
the  colored  rays,  which  together  make  up  white  light, 
concur  in  their  action  on  the  retina  at  a  given  moment ; 
in  ordinary  cases  it  is  immaterial  whether  these  rays 
have  left  the  luminous  body  successively  or  together. 
But  it  is  otherwise  when  a  luminous  body  becomes  visi- 
ble suddenly,  as  in  the  case  of  the  satellites  of  Jupiter, 
or  Saturn,  after  their  eclipses.  At  certain  periods,  more 
than  forty-nine  minutes  are  requisite  for  the  transmis- 
sion of  light  from  Jupiter  to  the  earth.  Now,  at  the 
moment  when  one  of  Jupiter's  satellites,  which  has 
been  eclipsed  by  that  planet,  emerges  from  the  shadow, 
the  red  rays,  if  their  velocity  were  the  greatest,  would 
evidently  reach  the  eye  first,  the  orange  next,  and  so  on 
through  the  chromatic  scale,  until  finally  the  comple-* 
ment  of  colors  would  be  filled  by  the  arrival  of  the  vio- 
let ray,  whose  velocity  is  supposed  to  be  the  least.  The 
satellite,  immediately  after  its  emersion,  would  appear 

*  Silliman's  Journal,  2d  series,  vol.  vii,  p.  364  seq. 


96  CONCEPTS  OF   MODERN   PHYSICS. 

red,  and  gradually,  in  proportion  to  the  arrival  of  the 
other  rays,  pass  into  white.  Conversely,  at  the  begin- 
ning of  the  eclipse,  the  violet  rays  would  continue  to 
arrive  after  the  red  and  other  intervening  rays,  and  the 
satellite,  up  to  the  moment  of  its  total  disappearance, 
would  gradually  shade  into  violet. 

Unfortunately  for  Cauchy's  hypothesis,  the  most 
careful  observation  of  the  eclipses  in  question  has  failed 
to  reveal  any  such  variations  of  color,  either  before  im- 
mersion or  after  emersion,  the  transition  between  light 
and  darkness  taking  place  instantaneously  and  without 
chromatic  gradations. 

Astronomy  points  to  several  other  phenomena  which 
are  equally  at  war  with  the  doctrine  of  unequal  veloci- 
ties in  the  movements  of  the  chromatic  undulations. 
Fixed  stars  beyond  the  parallactic  limit,  whose  light 
must  travel  more  than  three  years  before  it  reaches  us, 
are  subject  to  great  periodical  variations  of  splendor ; 
and  yet  these  variations  are  unaccompanied  by  varia- 
tions of  color.  Again,  the  assumption  of  different 
velocities  for  the  different  chromatic  rays  is  discoun- 
tenanced by  the  theory  of  aberration.  Aberration  is 
due  to  the  fact  that,  in  all  cases  where  the  orbit  of  the 
planet,  on  which  the  observer  is  stationed,  forms  an 
angle  with  the  direction  of  the  luminar  ray,  a  composi- 
tion takes  place  between  the  motion  of  the  light  and  the 
motion  of  the  planet,  so  that  the  direction  in  which  the 
light  meets  the  eye  is  a  resultant  of  the  two  component 
directions — the  direction  of  the  ray  and  that  of  the  ob- 
^erver's  motion.  If  the  several  rays  of  color  moved 
with  different  velocities  there  would  evidently  be  sev- 
eral resultants,  and  each  star  would  appear  as  a  colored 
spectrum  longitudinally  parallel  to  the  direction  of  the 
earth's  motion. 


THE  ATOMIC  THEORY.  97 

The  allegation  of  a  dependence  of  the  velocity  of 
the  undulatory  movements,  which  correspond  to,  or 
produce,  the  different  colors,  upon  the  length  of  the 
waves,  is  thus  at  variance  with  observed  fact.  The 
hypothesis  of  "  finite  intervals  "  is  unavailable  as  a  sup- 
plement to  the  undulatory  theory ;  other  methods  will 
have  to  be  resorted  to  in  order  to  free  this  theory  from 
its  difficulties.* 

The  negative  evidence  here  adduced  against  the 
supposition  of  an  atomic  or  molecular  constitution  of 
the  light-bearing  medium  is  reenforced  by  positive  evi- 
dence derived  from  a  branch  of  the  atomic  theory  itself 
— the  modern  science  of  thermo-dynamics.  Maxwell 
has  remarked,  with  obvious  truth,  that  such  a  medium 
(whose  atoms  or  molecules  are  supposed  to  penetrate 
the  intermolecular  spaces  of  ordinary  substances)  would 
be  nothing  more  nor  less  than  a  gas,  though  a  gas  of 

*  Since  the  publication  of  Cauchy's  "  Memoire  sur  la  dispersion  de  la 
lumiere  "  (Prag,  1836),  the  dependence  of  the  dispersive  powers  of  dif- 
ferent substances  upon  their  states  of  aggregation  and  chemical  composi- 
tion has  been  the  subject  of  extensive  experimental  research ;  and  the 
most  prominent  physicists  (Briot,  Holtzmann,  Redtenbacher,  C.  Neumann, 
Ketteler)  now  look  for  an  explanation  of  the  phenomena  of  dispersion  to 
the  action  of  ponderable  matter,  or  to  the  interaction  between  it  and  the 
aether.  Cf .  Briot,  "  Essai  sur  la  theorie  mathematique  de  la  lumiere  " 
(Paris,  Mallet-Bachelier,  1864),  p.  89  seq.  ;  Redtenbacher,  "  Dynamiden- 
system,"  p.  130  seq. ;  Ketteler,  "  Ueber  den  Einfluss  der  ponderablen 
Molekuele  auf  die  Dispersion  des  Lichts,"  etc.  (Pogg.  Ann.,  vol.  cxl, 
pp.  2  seq.  and  177  seq.  An  electro-magnetic  theory  of  light,  suggested 
by  the  proximate  equality  of  the  velocities  with  which  light  and  electro- 
magnetic disturbances  appear  to  be  propagated  through  air  and  other 
media,  and  by  the  action  of  a  magnet  (observed  by  Faraday)  in  turning 
the  plane  of  polarization  round  the  direction  of  the  luminar  ray  as  an 
axis,  was  broached  by  Clerk  Maxwell  in  1865,  and  has  recently  been  set 
forth  at  some  length  in  his  "  Treatise  on  Electricity  and  Magnetism,"  vol. 
ii,  pp.  383  seq.  This  theory  is  now  being  developed  by  Helmholtz,  Lorentz, 
Fitzgerald,  J.  J.  Thomson,  and  Lord  Rayleigh. 


98  CONCEPTS  OF  MODERN  PHYSICS. 

great  tenuity,  and  that  every  so-called  vacuum  would 
in  fact  be  full  of  this  rare  gas  at  the  observed  temper- 
ature and  at  the  enormous  pressure  which  the  aether, 
in  view  of  the  functions  assigned  to  it  by  the  undulatory 
theories,  must  be  assumed  to  exert.  Such  a  gas,  there- 
fore, must  have  a  correspondingly  enormous  specific 
heat  equal  to  that  of  any  other  gas  at  the  same  tem- 
perature and  pressure,  so  that  the  specific  heat  of  every 
vacuum  would  be  incomparably  greater  than  that  of 
the  same  space  filled  with  any  other  known  gas.  This 
remarkable  consequence  is  not  only  without  experi- 
mental warrant,  but — inasmuch  as  it  would  apply  to  all 
vacua,  including  the  intermolecular  spaces  of  ordinary 
bodies  of  whatever  state  of  aggregation — is  in  effect  a 
fatal  aggravation  of  a  peculiar  difficulty  of  the  molecular 
theory  which  is  in  itself  formidable  to  the  highest  de- 
gree. In  the  third  chapter  *  I  have  adverted  to  the  fact 
that,  when  a  body  is  heated,  a  part  only  of  the  energy 
communicated  to  it  appears  in  the  form  of  temperature, 
i.  e.  (in  the  sense  of  modern  theories),  of  progressive 
motions  of  the  molecules,  the  other  part  being  expended 
in  the  production  of  vibratory  or  rotatory  motions  of 
their  constituent  elements.  According  to  the  kinetic 
theory  of  gases,  this  latter  part,  the  internal  energy,  so 
called,  increases  with  the  number  of  variables  or  degrees 
of  freedom  in  each  molecule,  and  with  it,  therefore,  the 
specific  heat,  i.  e.,  the  ratio  of  the  whole  energy  to  that 
of  translation  which  produces  expansion  or  pressure, 
and  is  thus  exhibited  as  temperature.  If  the  mol- 
ecules were  "material  points"  without  internal  mo- 
bility, or  perfectly  elastic  and  perfectly  smooth  spheres, 
the  total  energy  would  be  available  for  the  production 
of  translatory  motion,  and  no  part  of  it  would  be  con- 

*  Supra,  p.  36. 


THE   ATOMIC   THEORY.  99 

verted  into  internal  energy.  But  if  the  molecules, 
though  perfectly  elastic,  are  not  perfect  spheres — as 
they  can  not  be,  whenever  they  consist  of  several  atoms 
each — the  specific  heat  must  at  least  be  equal  to  a  cer- 
tain minimum  assigned  by  the  theory.  Now,  the  spe- 
cific heats  of  oxygen,  nitrogen,  and  hydrogen  (all  which 
are  diatomic,  their  molecules  consisting  of  at  least  two 
atoms  each),  as  experimentally  ascertained  from  a  com- 
parison of  their  specific  heats  at  constant  pressure  and 
at  constant  volume,  fall  short  of  this  minimum.  And 
this  theoretical  minimum  would  be  very  materially  in- 
creased by  the  addition  of  the  specific  heat  due  to  the 
intermolecular  aether,  if  this  were  also  of  atomic  or 
molecular  constitution ;  the  discrepancy  between  the 
theoretical  postulates  and  the  experimental  data  would 
be  immeasurably  widened. 

3.  The  third  proposition  of  the  atomic  hypothesis 
assigns  to  the  atoms,  which  are  said  to  compose  the 
different  chemical  elements,  determinate  weights  cor- 
responding to  their  equivalents  of  combination,  and  is 
supposed  to  be  necessary  to  account  for  the  facts  whose 
enumeration  and  discussion  constitute  the  science  of 
chemistry.  The  proper  verification  of  these  facts  is  of 
great  difficulty,  because  they  have  generally  been  ob- 
served through  the  lenses  of  the  atomic  theory,  and 
stated  in  its  doctrinal  terms.  Thus  the  differentiation 
and  integration  of  bodies  are  invariably  described  as  de- 
composition and  composition  ;  the  equivalents  of  com- 
bination are  designated  as  atomic  weights  or  volumes, 
and  the  greater  part  of  chemical  nomenclature  is  a  sys- 
tematic reproduction  of  the  assumptions  of  atomism. 
Nearly  all  the  facts  to  be  verified  are  in  need  of  pre- 
paratory enucleation  from  the  envelopes  of  this  theory. 

The  phenomena  usually  described  as  chemical  coin- 


100  CONCEPTS  OF  MODERN   PHYSICS. 

position  and  decomposition  present  themselves  to  obser- 
vation thus :  A  number  of  heterogeneous  bodies  concur 
in  definite  proportions  of  weight  or  volume ;  they  inter- 
act ;  they  disappear,  and  give  rise  to  a  new  body  pos- 
sessing properties  which  are  neither  the  sum  nor  the 
mean  of  the  properties  of  the  bodies  concurring  and 
interacting  (excepting  the  weight  which  is  the  aggre- 
gate of  the  weights  of  the  interacting  bodies)  ;  and  this 
conversion  of  several  bodies  into  one  is  accompanied,  in 
most  cases,  by  changes  of  volume,  and  in  all  cases  by 
the  evolution  or  involution  of  heat,  or  other  forms  of 
energy.  Conversely,  a  single  homogeneous  body  gives 
rise  to  heterogeneous  bodies,  between  which  and  the 
body  from  which  they  originate  the  persistence  of 
weight  is  the  only  relation  of  identity. 

For  the  sake  of  convenience,  these  phenomena  may 
be  distributed  into  three  classes,  of  which  the  first  em- 
braces the  persistence  of  weight  and  the  combination  in 
definite  proportions ;  the  second,  the  changes  of  volume 
and  the  evolution  or  involution  of  energy ;  and  the  third, 
the  emergence  of  a  wholly  new  complement  of  chemical 
properties. 

17  Obviously,  the  atomic  hypothesis  is  in  no  sense  an 
explanation  of  the  phenomena  of  the  second  class.  It 
is  clearly  and  confessedly  incompetent  to  account  for 
changes  of  volume,  temperature,  or  latent  energy.  And, 
with  the  phenomena  of  the  third  class,  it  is  apparently 
incompatible.  For,  in  the  light  of  the  atomic  hypothesis, 
chemical  compositions  and  decompositions  are  in  their 
nature  nothing  more  than  aggregations  and  segregations 
of  masses  whose  integrity  remains  inviolate.  But  the 
radical  change  of  chemical  properties,  which  is  the 
result  of  all  true  chemical  action,  and  serves  to  distin- 
guish it  from  mere  mechanical  mixture  or  separation, 


THE   ATOMIC  THEORY. 

evinces  a  thorough  destruction  of  that  integrity.  It  may 
be  that  the  appearance  of  this  incompatibility  can  be 
obliterated  by  the  device  of  ancillary  hypotheses  ;  but 
that  leads  to  an  abandonment  of  the  simplicity  of  the 
atomic  hypothesis  itself,  and  thus  to  a  surrender  of  its 
claims  to  merit  as  a  theory.  7/ 

At  best,  then,  the  hypothesis  of  atoms  of  definite 
and  different  weights  can  be  offqr^d^^  &xplai^lon 
of  the  phenomena  of  the  first  class.  JDoes  it  explain 
them  in  the  sense  of  generalizing^- ,tljiem,;  of  T^ucing 
many  facts  to  one  ?  Not  at  all ;  it  accounts  for  them, 
as  it  professed  to  account  for  the  indestructibility  and 
impenetrability  of  matter,  by  simply  iterating  the  ob- 
served fact  in  the  form  of  an  hypothesis.  It  is  another 
case  (to  boiTOw  a  scholastic  phrase)  of  illustrating  idem 
per  idem.  It  says :  The  large  masses  combine  in  defi- 
nitely-proportionate weights  because  the  small  masses, 
the  atoms  of  which  they  are  multiples,  are  of  definitely- 
proportionate  weight.  It  pulverizes  the  fact,  and  claims 
thereby  to  have  sublimated  it  into  a  theory.* 

The  truth  is,  as  Sir  William  Thomson  has  observed, 
that  "  the  assumption  of  atoms  can  explain  no  property 
of  a  body  which  has  not  previously  been  attributed  to 
the  atoms  themselves." 

The  foregoing  considerations  do  not,  of  course,  de- 
tract from  the  merits  of  the  atomic  hypothesis  as  a 
graphic  or  expository  device — as  an  aid  to  the  repre- 

*  That  the  assumption  of  atoms  of  different  specific  gravities  is,  on 
the  basis  of  the  atomic  theory  itself,  simply  absurd,  has  already  been 
shown  (supra,  p.  28).  According  to  the  mechanical  conception,  which 
underlies  the  whole  atomic  hypothesis,  differences  of  weight  are  differ- 
ences of  density ;  and  differences  of  density  are  differences  of  distance 
between  the  particles  contained  in  a  given  space.  But,  in  the  atom  there 
is  no  multiplicity  of  particles  and  no  void  space ;  hence  differences  of 
density  or  weight  are  impossible  in  the  case  of  atoms. 


102  CONCEPTS  OF  MODERN  PHYSICS. 

sentative  faculty  in  "  realizing  "  the  phases  of  chemical 
or  physical  transformation.  It  is  a  fact  beyond  dispute 
that  chemistry  owes  a  great  part  of  its  practical  advance 
to  its  use,  and  that  the  structural  formulae  founded 
upon  it  have  enabled  the  chemist,  not  merely  to  trace 
the  connection  and  mutual  dependence  of  the  various 
stasres  in  the  metamorphosis  of  "  elements  "  and  "  com- 

*  *7 

fiwfej"  so'-palle^,;  tui  in  many  cases  (such  as  that  of 
.  hydrocarbon  'series  ,  in  organic  chemistry)  success- 
'jt^«aitjcij)(itelth.^  results  of  experimental  research. 
The  question,  to  wnat  extent  the  atomic  theory  is  still 
indispensable  to  the  chemist  as  a  "  working  hypothesis," 
is  at  this  moment  under  vigorous  discussion  among  men 
of  the  highest  scientific  authority,  many  of  whom  do 
not  hesitate  to  indorse  the  declaration  of  Cournot  (made 
many  years  ago)  that  "  the  belief  in  atoms  is  rather  a 
hindrance  than  a  help  "  *  not  only  because,  as  Cournot 
complains,  it  interposes  an  impassable  chasm  between 
the  phenomena  of  the  inorganic  and  those  of  the  organic 
world,  but  because  even  as  a  representation  of  the  phases 
and  results  of  the  most  ordinary  chemical  processes  it  is 
both  inadequate  and  misleading.  The  modifications  to 
which  it  has  lately  been  found  necessary  to  subject  it, 
in  order  to  meet  the  exigencies  of  the  present  state  of 
chemical  science — modifications  exemplified  in  the  doc- 
trines of  constant  and  varying  atomicities  or  valences,  of 
molecular  or  atomic  enchainments,  etc.,  with  the  attend- 
ant theories  (propounded  by  Kekule  and  others)  of  mo- 
lecular impact — attest  the  difficulties  encountered  in  the 

*  "En  somme,  pour  Pharmonie  generate  du  systeme  de  nos  connais- 
sances,  par  consequent  (autant  que  nous  pouvons  en  juger)  pour  la  plus 
juste  perception  de  1'harmonie  qui  certainement  existe  dans  1'ensemble 
des  choses,  la  foi  dans  les  atomes  est  plutot  un  embarras  qu'un  secours." 
Cournot,  Traite  de  PEnchainement  des  Idees  Fondamentales  dans  les 
Sciences  et  dans  PHistoire,  i,  264  seq. 


THE  ATOMIC   THEORY. 

attempt  to  bring  the  atomic  hypothesis  into  conformity 
with  the  theoretical  requirements  of  the  hour.  And,  in 
proportion  as  the  attention  of  the  modern  chemist  is 
directed  to  the  transference  and  transformation  of  en- 
ergy involved  in  every  instance  of  chemical  "  composi- 
tion "  and  "  decomposition  "  no  less  than  in  every  case  of 
allotropic  change,  its  ineptitude  as.  a  figurative  adumbra- 
tion of  the  real  nature  of  chemical  processes  becomes 
more  and  more  apparent.* 

I  propose  next  to  consider  one  of  the  most  notable 
applications  of  the  atomic  hypothesis  to  physics — the 
kinetic  theory  of  gases. 

*  As  an  illustration  of  the  disfavor  with  which  the  atomic  hypothesis 
is  coming  to  be  regarded  by  distinguished  chemists,  I  may  be  permitted  to 
quote  a  passage  from  an  essay  by  the  late  Sir  Benjamin  C.  Brodie,  Pro- 
fessor of  Chemistry  at  Oxford :  "  I  can  not  but  say  that  I  think  the  atomic 
doctrine  has  proved  itself  inadequate  to  deal  with  the  complicated  system 
of  chemical  fact  which  has  been  brought  to  light  by  the  efforts  of  modern 
chemists.  I  do  not  think  that  the  atomic  theory  has  succeeded  in  con- 
structing an  adequate,  a  worthy,  or  even  a  useful  representation  of  those 
facts."  "  On  the  Mode  of  Representation  afforded  by  the  Chemical  Calcu- 
lus as  contrasted  with  the  Atomic  Theory."  Chemical  News,  August, 
1867,  p.  72.  It  is  but  fair  to  add,  however,  that  I  am  not  in  sympathy 
with  Brodie's  own  theoretical  scheme  so  far  as  I  understand  it. 


CHAPTER  VIII. 

THE   KINETIC   THEORY   OF   GASES. CONDITIONS    OF  THE 

VALIDITY   OF   SCIENTIFIC    HYPOTHESES. 

IN  the  fourth  chapter  *  I  have  already  given  an  out- 
line of  the  doctrine  now  generally  known  and  accepted 
as  the  kinetic  theory  of  gases.  The  assumptions  of 
this  theory  are  that  a  gaseous  body  consists  of  a  great 
number  of  minute  solid  particles — molecules  or  atoms 
— in  perpetual  rectilinear  motion,  which,  as  a  whole,  is 
conserved  by  reason  of  the  absolute  elasticity  of  the 
moving  particles,  while  the  directions  of  the  move- 
ments of  the  individual  particles  are  incessantly  changed 
by  their  mutual  encounters  or  collisions.  The  colliding 
particles  are  supposed  to  act  upon  each  other  only  with- 
in very  small  distances  and  for  very  short  times  before 
and  after  collision,  their  motion  being  free,  and  conse- 
quently rectilinear,  in  the  intervals  between  such  dis- 
tances and  times.  The  durations  of  the  rectilinear 
motions  in  free  paths  are,  moreover,  assumed  to  be  in- 
definitely large  as  compared  with  the  durations  of  the 
encounters  and  of  the  mutual  actions. 

This  theory  was  first  advanced  by  Kroenig,  f  and 
has  since  been  elaborated  by  Clausius,  Maxwell,  Boltz- 

*  Supra,  p.  40. 

f  Pogg.  Ann.,  vol.  xcix,  p.  315  seq.  As  is  usual  in  such  cases,  pro- 
lusions of  the  theory  have  since  been  discovered  in  the  writings  of  vari- 
ous older  physicists — cf.  P.  Du  Bois-Reymond  in  Pogg.  Ann.,  vol.  cvii, 
p.  490  seq. 


THE  KINETIC  THEORY   OF  GASES.  1Q5 

maim,  Stefan,  Pfaundler,  and  other  physicists  of  the 
highest  note.  As  in  the  case  of  the  atomic  hypothesis 
generally,  I  propose  for  the  present  to  discuss,  not  so 
much  the  logical  warrant,  as  the  scientific  value,  of  the 
theory  in  question.  To  this  end  it  will  be  necessary, 
however,  first  to  ascertain  the  true  nature  and  function 
of  a  scientific  hypothesis — not  only  the  criteria  of  its 
value,  but  also  the  conditions  of  its  validity. 

[A  scientific  hypothesis  may  be  defined  in  general 
terms  as  a  provisional  or  tentative  explanation  of  phys- 
ical phenomena.*  But  what  is  an  explanation  in  the 
true  scientific  sense  ?  The  answers  to  this  question 
which  are  given  by  logicians  and  men  of  scien  ce,  though 
differing  in  their  phraseology,  are  essentially  of  the 
same  import.  Phenomena  are  explained  by  an  exhibi- 
tion of  their  partial  or  total  identity  with  other  phe- 
nomena. Science  is  knowledge ;  and  all  knowledge,  in 
the  language  of  Sir  William  Hamilton,  f  is  a  "  unifica- 
tion of  the  multiple."  "  The  basis  of  all  scientific 
explanation,"  says  Bain,  J  "  consists  in  assimilating  a 
fact  to  some  other  fact  or  facts.  It  is  identical  with 
the  generalizing  process."  And  "  generalization  is  only 
the  apprehension  of  the  One  in  the  Many."  *  Similar- 
ly Jevons  :  [  "  Science  arises-  from  the  discovery  of 
identity  amid  diversity,"  and  A  "  every  great  advance 
in  science  consists  in  a  great  generalization  pointing 

*  Wundt  has  lately  (Logik,  i,  403)  sought  to  distinguish  hypothe- 
ses from  "  anticipations  of  fact  "  and  to  restrict  the  term  "  hypothesis  " 
to  a  sense  which,  notwithstanding  its  etymological  warrant,  is  at  variance 
with  ordinary  as  well  as  scientific  usage. 

f  Lectures  on  Metaphysics  (Boston  ed.),  pp.  47,  48. 
\  Logic,  ii  (Inductive),  chap,  xii,  §  2. 

*  Hamilton,  1.  c.,  p.  48. 

I  Principles  of  Science,  i,  p.  1. 
A  76.,  ii,  p.  281. 


106  CONCEPTS  OF  MODERN  PHYSICS. 

out  deep  and  subtle  resemblances."  The  same  thing  is 
stated  by  the  author  just  quoted  in  another  place :  * 
"  Every  act  of  explanation  consists  in  detecting  and 
pointing  out  a  resemblance  between  facts,  or  in  showing 
that  a  greater  or  less  degree  of  identity  exists  between 
apparently  diverse  phenomena." 

All  this  may  be  expressed  in  familiar  language  thus : 
When  a  new  phenomenon  presents  itself  to  the  man  of 
science  or  to  the  ordinary  observer,  the  question  arises 
in  the  mind  of  either  :  What  is  it  ?— and  this  question 
simply  means :  Of  what  known,  familiar  fact  is  this 
apparently  strange,  hitherto  unknown  fact  a  new  pres- 
entation—of what  known,  familiar  fact  or  facts  is  it  a 
disguise  or  complication  ?  Or,  inasmuch  as  the  par- 
tial or  total  identity  of  several  phenomena  is  the  basis 
of  classification  (a  class  being  a  number  of  objects  hav- 
ing one  or  more  properties  in  common)Kit  mayj  also  [be 

/    said  that  all  explanation,  including  explanation  by  hy- 
pothesis, is  in  its  nature  classification. 

Such  being  the  essential  nature  of  a  scientific  ex- 
planation of  which  an  hypothesis  is  a  probatory  form, 
it  follows  that  no  hypothesis  can  be  valid  which  does 
not  identify  the  whole  or  a  part  of  the  phenomenon, 
for  the  explanation  of  which  it  is  advanced,  with  some 
other  phenomenon  or  phenomena  previously  observed. 
This  first  and  fundamental  canon  of  all  hypothetical 
reasoning  in  science  is  formally  resolvable  into  two 

i  propositions,  the  first  of  which  is  that  every  valid  hy- 
pothesis must  be  an  identification  of  two  terms — the 
fact  to  be  explained  and  a  fact  by  which  it  is  explained ; 
and  the  second  that  the  latter  fact  must  be  known  to 
experience. 

Tested  by  the  first  of  these  propositions,  all  hypoth- 

*  Principles  of  Science,  ii,  p.  166. 


THE  KINETIC  THEORY  OF  GASES.  107 

eses  are  futile  which  merely  substitute  an  assumption 
for  a  fact,  and  thus,  in  the  language  of  the  schoolmen, 
explain  obscurum  per  obscurius,  or  (the  assumption  be- 
ing simply  the  statement  of  the  fact  itself  in  another 
form — the  "  fact  over  again")  illustrate  idem  per  idem. 
And  the  futility  of  such  hypotheses  goes  to  the  verge 
of  mischievous  puerility  when  they  replace  a  single 
fact  by  a  number  of  arbitrary  assumptions,  among  which 
is  the  fact  itself.  Some  of  the  uses  made  of  the  atomic 
hypothesis,  both  in  physics  and  chemistry,  which  have 
been  discussed  in  the  last  chapter,  afford  conspicuous 
examples  of  this  class  of  bootless  assumptions ;  and 
similar  instances  abound  among  the  mathematical  for- 
mulae that  are  not  infrequently  paraded  as  physical 
theories.  These  formulae  are  in  many  cases  simply  re- 
sults of  a  series  of  transformations  of  an  equation 
which  embodies  an  hypothesis  whose  elements  are 
neither  more  nor  less  than  the  elements  of  the  phe- 
nomenon to  be  accounted  for,  the  sole  merit  of  the 
emerging  formula  beijig  that  it  is  not  in  conflict  with 
the  initial  one.* 

*  I  hope  not  to  be  misunderstood  as  disparaging  the  services  for 
which  physical  science  is  indebted  to  mathematics.  These  services — es- 
pecially those  rendered  by  modern  analysis — are  incalculable.  But  there 
are  mathematicians  who  imagine  that  they  have  compassed  a  solution  of 
all  the  mysteries  involved  in  a  case  of  physical  action  when  they  have 
reduced  it  to  the  form  of  a  differential  expression  preceded  by  a  group 
of  integral  signs.  Even  when  their  equations  are  integrable  they  should 
bear  in  mind  that  the  operations  of  mathematics  are  essentially  deduc- 
tive, and,  while  they  may  extend,  can  never  deepen  a  physical  theory. 
Granting  that  mathematics  are  much  more  than  Ka^dpfiara  ^vx^y>  and tnat 
their  office  in  the  investigation  of  the  causes  of  natural  phenomena  is 
far  more  important  than  the  purely  regulative  functions  of  formal  logic 
in  science  generally — conceding  that  the  application  of  mathematics  to 
physics  has  not  only  brought  to  light  the  significance  of  many  experi- 
mental results,  but  has  often  been  a  trustworthy  guide  to  successful  re- 


108  CONCEPTS   OF  MODERN   PHYSICS. 

In  order  to  comply  with  the  first  condition  of  its 
validity,  an  hypothesis  must  bring  the  fact  to  be  ex- 
plained into  relation  with  some  other  fact  or  facts  by 
identifying  the  whole  or  a  part  of  the  former  with  the 
whole  or  a  part  of  the  latter.  In  this  sense  it  has  been 
well  said  that  a  valid  hypothesis  reduces  the  number  of 
the  uncomprehended  elements  of  a  phenomenon  by  at 
least  one.*  In  the  same  sense  it  is  sometimes  said  that 

search — nevertheless  some  of  our  prominent  physicists  and  mathemati- 
cians might  still  read  with  profit  the  ninety-sixth  aphorism  in  the  first 
book  of  Bacon's  Novum  Organon :  "  Naturalis  Philosophia  adhuc  sin- 
cera  non  invenitur,  sed  infecta  et  corrupta ;  in  Aristotelis  schola  per 
logicam ;  in  Platonis  schola  per  theologiam  naturalem  ;  in  secunda  schola 
Platonis,  Prochi  et  aliorum  per  Mathematicam,  quae  philosophiam  natu- 
ralem terminare,  non  gcnerare  aut  procrcarc  dcbet"  As  to  the  value  of 
the  class  of  formulae  referred  to  in  the  text  it  may  not  be  inappropri- 
ate to  cite  the  words  of  Cournot  (De  1'Enchainement,  etc.,  i,  p.  249} : 
"  Tant  qu'un  calcul  ne  fait  que  rendre  ce  que  1'on  a  tire  de  1'observation 
pour  1'introduire  dans  les  elements  du  calcul  a  vrai  dire  il  n'ajoute  rien 
aux  donn6es  de  1'obserTation."  To  the  same  effect  are  the  admirable 
reflections  of  M.  Poinsot  (Theorie  Nouvelle  de  la  Rotation  des  Corps, 
ed.  1851,  p.  79) :  "  Ce  qui  a  pu  faire  illusion  a  quelques  esprits  sur  cette 
espece  de  force  qu'ils  supposent  aux  formules  de  1'analyse,  c'est  qu'on 
en  retire,  avec  assez  de  facilite,  des  verites  deja  connues,  et  qu'on  y  a, 
pour  ainsi  dire,  soi-meme  introduites,  et  il  semble  alors  que  1'analyse 
nous  donne  ce  qu'elle  ne  fait  que  nous  rendre  dans  un  autre  langage. 
Quand  un  the'oreme  est  connu,  on  n'a  qu'a  1'exprimcr  par  des  equations  ; 
si  le  theoreme  est  vrai,  chacune  d'elle  ne  peut  manquer  d'etre  exacte, 
aussi  bien  que  les  transformers  qu'on  en  peut  deduire  ;  et  si  Ton  arrive 
ainsi  a  quelque  formule  evidente,  ou  bien  etablie  d'ailleurs,  on  n'a  qu'a 
prcndre  cette  expression  comme  un  point  de  depart,  a  revenir  sur  ses  pas, 
et  le  calcul  seul  parait  avoir  conduit  comme  de  lui-meme  au  theoreme 
dont  il  s'agit.  Mais  c'est  en  cela  que  le  lecteur  est  trorape." 

*  "  Der  Verstand  hat  das  Beduerf  niss  jede  Erscheinung  zu  erklaeren 
d.  h.  dieselbe  als  das  Resultat  bekannter  Kraef  te  oder  Erscheinungen  be- 
grifflich  abzuleiten.  .  .  .  Es  geht  hieraus  hervor,  dass  jede  Hypothese  nur 
bekannte  Kraefte  oder  Erscheinungen  zur  Erklaerung  annehmen  darf,  in- 
dem  die  Annahme  einer  bisher  unbekannten  Kraft  nur  die  Qualitaet  des 
zu  erklaerenden  Phaenomen's  aendern,  aber  nicht  die  Zahl  der  unerklaer- 
ten  Momente  reduciren  kann.  Soil  cine  Hypothese  nicht  vollkommen  un- 


THE  KINETIC  THEORY   OF  GASES.  1Q9 

every  true  theory  or  hypothesis  is  in  effect  a  simplifica- 
tion of  the  data  of  experience — an  assertion  which  must 
be  understood,  however,  with  due  regard  to  the  second 
proposition  to  be  discussed  presently,  i.  e.,  with  the 
proviso  that  the  theory  be  not  a  mere  asylum  ignorarv- 
tice,  of  the  kind  denoted  by  the  schoolmen  as  a  prin- 
cipium  expressivum,  such  as  the  explanation  of  the 
phenomena  of  life  by  reference  to  a  vital  principle ',  or 
of  certain  chemical  processes  by  catalytic  action.  True 
scientific  explanations  are  generally  complicated  in  form, 
not  only  because  most  phenomena,  on  proper  analysis, 
prove  to  be  complex,  but  because  the  simplest  fact  is 
not  the  effect  of  a  single  cause,  but  the  product  of  a 
great  and  often  indeterminate  multiplicity  of  agencies 
— the  outcome  of  the  concurrence  of  numerous  condi- 
tions. The  Newtonian  theory  of  planetary  motion  is 
much  more  intricate  than  that  of  Kepler,  according  to 
which  every  planet  is  conducted  along  its  path  by  an 
angelus  rector  ;  and  the  account  given  by  modern  celes- 
tial mechanics  of  the  precession  of  the  equinoxes  is  far 
less  simple  than  the  announcement  that  among  the  great 
periods  originally  established  by  the  Author  of  the  uni- 
verse was  the  Hipparchian  cycle.  The  old  brocard, 
simplex  veri  judicium,  is  to  be  taken  with  many  grains 
of  allowance  before  it  can  be  trusted  as  a  safe  rule  in 
determining  the  validity  or  value  of  scientific  doctrines. 
I  now  come  to  the  second  requirement  of  the  valid- 
ity of  an  hypothesis  :  that  the  explanatory  phenomenon 
(i.  e.,  that  with  which  the  phenomenon  to  be  explained 
is  identified)  must  be  a  datum  of  experience.  This 

nuetz  und  demgemaess  die  Verstandesarbeit,  welche  sie  zur  Befriedigung 
eines  Beduerf nisses  erzeugte,  keine  zwecklose  sein,  so  muss  jede  Hypothese 
die  ZaM  der  unbegriffenen  Mbmente  einer  Erschdnung  mindestens  urn  eine 
crniedrigen."  Zoellner,  Natur  der  Kometen,  p.  189  seq. 


CONCEPTS  OF  MODERN  PHYSICS. 

proposition  is  in  substance  equivalent  to  that  part  of 
Newton's  first  regula  philosophandi,*  in  which  he  in- 
sists that  the  cause  assigned  for  the  explanation  of  nat- 
ural things  must  be  a  vera  causa — a  term  which  he 
does  not  expressly  define  in  the  Principia,  but  whose 
import  may  be  gathered  from  the  following  passage 
of  his  Opticks f  :  "To  tell  us  that  every  species  of 
things  is  endowed  with  an  occult  specific  quality  by 
which  it  acts  and  produces  manifest  effects  is  to  tell  us 
nothing.  But  to  derive  two  or  three  general  principles 
of  motion  from  phenomena  and  afterward  to  tell  us  how 
the  properties  and  actions  of  all  corporeal  things  follow 
from  these  manifest  principles  would  be  a  very  great 
step  in  philosophy,  though  the  causes  of  those  principles 
were  not  yet  discovered." 

The  requirement  in  question  has  long  been  the  sub- 
ject of  animated  discussion  by  J.  S.  Mill,  Whewell,  and 
others ;  but  it  will  be  found,  I  think,  that,  after  making 
due  allowance  for  necessary  implications,  there  is  little 
real  disagreement  among  thinkers.  The  recent  state- 
ment of  G.  H.  Lewes  \  that  u  an  explanation  to  be 
valid  must  be  expressed  in  terms  of  phenomena  already 
observed,"  and  the  counter-statement  of  Jevons  *  that 
"  agreement  with  fact  (i.  e.,  the  fact  to  be  explained)  is 
the  one  sole  and  sufficient  test  of  a  true  hypothesis," 
are  both  far  too  broad,  and  are,  indeed,  modified  by 
Lewes  and  Jevons  themselves  in  the  progress  of  the 
.  discussion ;  but  the  claim  of  Mr.  Lewes  is  nevertheless 
true  in  the  sense  that  no  explanation  is  real  unless  it  is 
an  identification  of  experiential  data.  The  confusion 
which,  as  in  so  many  other  cases  of  scientific  contro- 
versy, is  at  the  bottom  of  the  seeming  disagreement 

*  Phil.  Nat.  Princ.  Math.,  lib.  iii.  f  Fourth  edition,  p.  377. 

\  Problems  of  Life  and  Mind,  ii,  7.         *  Princ.  of  Science,  ii,  138. 


THE  KINETIC  THEORY  OF  GASES. 

between  the  contending  parties,  arises  from  a  disregard 
of  the  circumstance  that  the  identification  of  two  phe- 
nomena may  be  both  partial  and  indirect — that  it  may 
be  effected  by  showing  that  the  phenomena  have  some 
known  feature  in  common  on  condition  that  the  exist- 
ence, in  one  or  both  of  the  phenomena,  of  some  other 
feature  not  yet  directly  observed,  and  perhaps  incapable 
of  direct  observation,  be  assumed.  The  aptest  illustra- 
tion of  this  is  the  much-debated  undulatory  theory  of 
light.  This  hypothesis  identifies  light  with  other  forms 
of  radiance,  and  even  with  sound,  by  showing  that  all 
these  phenomena  have  the  element  of  vibration  or 
undulation  (which  is  well  known  to  experience)  in 
common,  on  the  assumption  of  an  all-pervading  ma- 
terial medium,  of  a  kind  wholly  unknown  to  experi- 
ence, as  the  bearer  of  the  luminar  undulations.  In  this 
case,  as  in  all  similar  cases,  the  identity  lies,  not  in  the 
fictitious  element,  the  aether,  but  in  the  real  element, 
the  undulation.  It  consists,  not  in  the  agent •,  but  in 
the  law  of  its  action.  And  it  is  obvious  that  every 
hypothesis  which  establishes  coincidences  between  phe- 
nomena in  particulars  that  are  purely  fictitious  is  wholly 
vain,  because  it  is  in  no  sense  an  identification  of  phe- 
nomena. It  is  worse  than  vain:  it  is  meaningless — a 
mere  collection  of  words  or  symbols  without  compre- 
hensive import.  As  Jevons  expresses  it :  *  "  No  hy- 
pothesis can  be  so  much  as  framed  in  the  mind,  unless 
it  be  more  or  less  conformable  to  experience.  As  the 
material  of  our  ideas  is  undoubtedly  derived  from  sen- 
sation so  we  can  not  figure  to  ourselves  any  existence 
or  agent  but  as  endowed  with  some  of  the  properties 
of  matter.  All  that  the  mind  can  do  in  the  creation 
of  new  existences  is  to  alter  combinations,  or  by  anal- 

*  Princ.  of  Science,  ii,  141. 


112  CONCEPTS  OF  MODERN  PHYSICS. 

ogy  to  alter  the  intensity  of  sensuous  properties."  J. 
S.  Mill  is,  therefore,  clearly  wrong  when  he  says  *  that, 
"  an  hypothesis  being  a  mere  supposition,  there  can  be 
no  other  limits  to  hypotheses  than  those  of  the  human 
imagination,"  and  that  "  we  may,  if  we  please,  imagine, 
by  way  of  accounting  for  an  effect,  some  cause  of  a 
kind  utterly  unknown  and  acting  according  to  a  law 
altogether  fictitious."  The  unsoundness  of  the  latter 
part  of  this  proposition  is  evidently  felt  by  Mill  himself, 
for  he  adds  at  the  end  of  the  next  sentence  that  "  there 
is  probably  no  hypothesis  in  the  history  of  science  in 
which  both  the  agent  itself  and  the  law  of  its  operation 
were  fictitious."  There  certainly  is  no  such  hypothesis 
— at  least  none  which  has  in  any  way  subserved  the 
interests  of  science. 

An  hypothesis  may  involve  not  only  one  but  sev- 
eral fictitious  assumptions,  provided  they  bring  into  re- 
lief, or  point  to  the  probability,  or  at  least  possibility, 
of  an  agreement  between  phenomena  in  a  particular 
that  is  real  and  observable.  This  is  especially  legiti- 
mate when  the  agreement  thus  brought  to  light  is  not 
between  two,  but  a  greater  number  of  phenomena,  and 
still  more  so  when  the  agreement  is  not  merely  in  one 
but  in  "  several  real  particulars  between  diverse  phe- 
nomena, so  that,  in  the  language  of  Whewell,f  "  the  hy- 
potheses which  were  assumed  for  one  class  of  cases  are 
found  to  explain  another  of  a  different  nature — a  con- 
silience of  induction."  An  instance  of  this  is  afforded 
by  the  hypothesis  just  referred  to  of  the  luminiferous 
aether,  which  was  at  first  believed  also  to  explain  the 
retardation  of  comets.  But,  while  the  probability  of  the 
truth  of  an  hypothesis  is  in  direct  ratio  to  the  number 

*  Logic,  8th  ed.,  p.  394. 

f  History  of  the  Inductive  Sciences  (Am.  ed.),  ii,  186 


THE  KINETIC   THEORY   OF   GASES. 

of  phenomena  thus  brought  into  relation,  it  is  in  the 
inverse  ratio  of  the  number  of  such  fictions,  or,  more 
accurately,  its  improbability  increases  geometrically 
while  the  series  of  independent  fictions  expands  arith- 
metically.* This  finds  illustration  again  in  the  undu- 
latory  theory  of  light.  The  multitude  of  fictitious  as- 
sumptions embodied  in  this  hypothesis,  in  conjunction 
with  the  failure  of  the  consiliences  by  which  it  appeared 
at  first  to  be  distinguished,  can  hardly  be  looked  upon 
otherwise  than  as  a  standing  impeachment  of  its  validity 
in  its  present  form.  However  ready  we  may  be  to  ac- 
cede to  the  demands  of  the  theorist  when  he  asks  us  to 
grant  that  all  space  is  pervaded,  and  all  sensible  matter 
is  penetrated,  by  an  adamantine  solid  exerting  at  each 
point  in  space  an  elastic  force  1,148,000,000,000  times 
that  of  air  at  the  earth's  surface,  and  a  pressure  upon 

*  "  En  general,"  says  Cournot  (De  1'Enchainement,  etc.,  i,  103), 
"  une  theorie  scientifique  quelconque,  imaginee  pour  relier  un  certain 
nombre  de  faits  donnes  par  1'observation,  peut  etre  assimilee  a  la  courbe 
que  1'on  trace  d'apres  une  loi  geometrique,  en  s'imposant  la  condition  de 
la  faire  passer  par  un  certain  nombre  de  points  donnes  d'avance.  Le 
jugement  que  la  raison  porte  sur  la  valeur  intrinseque  de  cette  theorie 
est  un  jugement  probable,  une  induction  dont  la  probabilite  tient  d'une 
part  a  la  simplicity  de  la  formule  the"orique,  d'autre  part  au  nombre  des 
faits  ou  des  groupes  des  faits  qu'elle  relie,  le  meme  groupe  devant  com- 
prendre  tous  les  faits  qui  s'expliquent  deja  les  uns  par  les  autres,  inde- 
pendamment  de  1'hypothese  theorique.  S'ilfaut  compliquer  la  formule  d 
mesure  que  de  nouveaux  faits  se  revelent  d  ^observation  elle  devient  de 
moins  en  moins probable  en  tant  que  loi  de  la  Nature  ;  ce  n'est  bientot  plus 
qu'un  echaf  audage  artificiel  qui  croule  enfin  lorsque,  par  un  surcroit  de 
complication,  elle  perd  meme  1'utilite  d'un  syst£me  artificiel,  celle  d'aider 
le  travail  de  la  pensee  et  de  diriger  les  recherches.  Si  au  contraire  les 
faits  acquis  a  1'observation  post6rieurement  a  la  construction  de  Phy- 
pothese  sont  relies  par  elle  aussi  bien  que  les  faits  qui  ont  servi  a  la 
construire,  si  surtout  des  faits  prevus  comme  consequences  de  1'hypothese 
re9oivent  des  observations  posterieures  une  confirmation  eclatante,  la 
probabilite  de  1'hypothese  peut  aller  jusqu'a  ne  laisser  aucune  place  au 
doute  dans  un  esprit  eclaireV' 


CONCEPTS  OF  MODERN  PHYSICS. 

the  square  inch  of  17,000,000,000,000  pounds  *— a  solid 
which,  at  the  same  time,  wholly  eludes  our  senses,  is 
utterly  impalpable  and  offers  no  appreciable  resistance 
to  the  motions  of  ordinary  bodies — we  are  appalled 
when  we  are  told  that  the  alleged  existence  of  this  ad- 
amantine medium,  the  aether,  does  not,  after  all,  explain 
the  observed  irregularities  in  the  periods  of  comets ; 
that,  furthermore,  not  only  is  the  supposed  luininif erous 
aether  unavailable  as  a  medium  for  the  origination  and 
propagation  of  dielectric  phenomena,  so  that  for  these 
a  distinct  all-pervading  electrif erous  aether  must  be. as- 
sumed, f  but  that  it  is  very  questionable  whether  the  as- 
sumption of  a  single  aethereal  medium  is  competent  to 
account  for  all  the  known  facts  in  optics  (as,  for  in- 
stance, the  non-interference  of  two  rays  originally  polar- 
ized in  different  planes  when  they  have  been  brought 
to  the  same  plane  of  polarization,  and  certain  phenom- 
ena of  double  refraction,  in  view  of  which  it  is  neces- 
sary to  suppose  that  the  rigidity  of  the  medium  varies 
with  the  direction  of  the  strain — a  supposition  discoun- 
tenanced by  the  facts  relating  to  the  intensities  of  re- 
flected light),  and  that  for  the  adequate  explanation  of 
the  phenomena  of  light  it  is  "  necessary  to  consider 
what  we  term  the  aether  as  consisting  of  two  media, 
each  possessed  of  equal  and  enormous  self-repulsion  or 
elasticity,  and  both  existing  in  equal  quantities  through- 
out space,  whose  vibrations  take  place  in  perpendicular 
planes,  the  two  media  being  mutually  indifferent,  neither 
attracting  nor  repelling."  J  In  this  endless  superfeta- 

*  Cf.  Herschel,  Familiar  Lectures,  etc.,  p.  282  ;  F.  De  Wrede  (Presi- 
dent Royal  Academy  of  Sciences  in  Stockholm),  address,  Phil.  Mag.,  4th 
ser.,  vol.  xliv,  p.  82. 

f  W.  A.  Norton,  on  Molecular  Physics,  Phil.  Mag.,  4th  aer.,  vol.  xxiii, 
p.  193. 

\  Hudson,  on  Wave  Theories  of  Light,  Heat,  and  Electricity,  Phil. 


THE  KINETIC  THEORY  OF  GASES. 

tion  of  aethereal  media  upon  space  and  ordinary  matter, 
there  are  ominous  suggestions  of  the  three  kinds  of 
aethereal  substances  postulated  by  Leibnitz  and  Cartesius 
alike  as  a  basis  for  their  vortical  systems.  There  is  an 
impulsive  whirl  in  our  thoughts,  at  least,  when  we  are 
called  upon,  in  the  interests  of  the  received  form  of 
the  undulatory  theory,  not  only  to  reject  all  the  pre- 
sumptions arising  from  our  common  observation  and 
all  the  analogies  of  experience,  but  to  cumulate  hy- 
potheses and  aethers  indefinitely.  And  we  are  but  par- 
tially reassured  by  the  circumstance  that  the  theory  in 
question,  besides  accounting  for  the  phenomena  of  op- 
tics which  had  been  observed  at  the  time  of  its  pro- 
mulgation, has  the  great  merit  of  successful  prevision, 
having  led  to  the  prediction  of  a  number  of  facts  sub- 
sequently discovered.  These  predictions,  certainly,  have 
not  only  been  numerous,  but  several  of  them,  such  as 
Hamilton's  announcement  of  conical  refraction  (after- 
ward verified  by  Lloyd)  and  FresnePs  forecast  (from 

Mag.  (iv),  vol.  xliv,  p.  210  s'eq.  In  this  article  the  author  also  points 
out  the  crudeness  of  the  subsidiary  hypotheses  which  have  been 
framed  to  obviate  other  difficulties  of  the  undulatory  theory,  among 
which  are  those  discussed  in  the  last  chapter.  "  Waves  of  sound,"  he 
says,  "  in  our  atmosphere  are  10,000  times  as  long  as  the  waves  of  light, 
and  their  velocity  of  propagation  about  850,000  times  less,  and,  even 
when  air  has  been  raised  to  a  temperature  at  which  waves  of  red  light 
are  propagated  from  matter,  the  velocity  of  sound-waves  is  only  increased 
to  about  double  what  it  was  at  zero  centigrade.  Even  their  velocity 
through  glass  is  55,000  times  less  than  the  speed  of  the  aethereal  undu- 
lations, and  the  extreme  slowness  of  change  of  temperature  in  the  con- 
duction of  heat  (as  contrasted  with  the  rapidity  with  which  the  vibrations 
of  the  aether  exhaust  themselves,  becoming  insensible  almost  instantly 
when  the  action  of  the  existing  cause  ceases)  marks  distinctly  the 
essential  difference  between  molecular  and  aethereal  vibrations.  It  ap- 
pears to  me,  therefore,  a  very  crude  hypothesis  to  imagine  a  combina- 
tion of  aethereo-molecular  vibrations  as  accounting  for  the  very  minute 
difference  in  the  retardation  of  doubly  refracted  rays  in  crystals." 


11(3  CONCEPTS  OF  MODERN  PHYSICS. 

the  imaginaiy  form  of  an  algebraic  formula)  of  circular 
polarization  after  two  internal  reflections  in  a  rhomb, 
are  very  striking.  But,  although  anticipations  of  this 
sort  justly  serve  to  accredit  an  hypothesis,  they  are,  as 
Mill  has  shown,*  by  no  means  absolute  tests  of  their 
truth.  Using  the  word  "  cause  "  in  the  sense  in  which 
it  is  commonly  understood,  an  effect  may  be  due  to 
any  one  of  several  causes,  and  may,  therefore,  in 
many  cases  be  accounted  for  by  any  one  of  several 
conflicting  hypotheses,  as  becomes  evident  to  the  most 
cursory  glance  at  the  history  of  science.  When  an 
hypothesis  successfully  explains  a  number  of  phenomena 
with  reference  to  which  it  was  constructed,  it  is  not 
strange  that  it  should  also  explain  others  connected 
with  them  that  are  subsequently  discovered.  There  are 
few  discarded  physical  theories  that  could  not  boast  the 
prevision  of  phenomena  to  which  they  pointed  and 
which  were  afterward  observed ;  among  them  are  the 
one-fluid  theory  of  electricity  and  the  corpuscular  theory 
of  light. 

There  are,  of  course,  other  conditions  of  the  validity 
of  an  hypothesis  to  which  I  have  not  yet  adverted. 
^  Among  them  are  those*  specified  by  Sir  W.  Hamilton, 
Mill,  Bain,  and  others,  such  as  that  the  hypothesis  must 
not  be  contradictory  of  itself  or  in  conflict  with  the 
known  laws  of  nature  (which  latter  requirement  is,  how- 
ever, somewhat  doubtful,  inasmuch  as  the  laws  in  ques- 
tion may  be  incomplete  inductions  from  past  experience 
to  be  supplemented  by  the  very  elements  postulated  by 

*  Logic,  p.  356.  Long  before  Mill,  Leibnitz  observed  that  success 
in  explaining  (or  predicting)  facts  is  no  proof  of  the  validity  of  an  hy- 
pothesis, inasmuch  as  right  conclusions  may  be  drawn  from  wrong  prem- 
isses— as  Leibnitz  expresses  it,  "comme  le  vrai  peut  etre  tire  du  faux." 
Cf.  Nouveaux  Essais,  chap,  xvii,  sec.  5 — Leibnitii,  opp.  ed.  Erdmann,  p. 
397. 


THE   KINETIC   THEORY   OF  GASES.  Il7 

the  hypothesis) ;  that  it  must  be  of  a  nature  to  admit  of 
deductive  inferences,  etc.  Upon  all  these  it  is  not  nec- 
essary, in  view  of  my  present  purpose,  to  dilate.  The 
two  conditions  which  I  have  sought  to  enforce  and  illus- 
trate are,  in  my  judgment,  sufficient  tests  of  the  validity 
and  merits  of  the  kinetic  theory  of  gases. 

The  fundamental  fact  to  be  accounted  for  by  this 
theory  is  that  gases  are  bodies  which,  at  constant  tem- 
peratures and  in  the  absence  of  external  pressure,  ex- 
pand at  even  rate.  From  this  fact  the  two  great  empir- 
ical laws,  so  called,  expressive  of  those  physical  proper- 
ties of  a  gas  which  are  directly  attested  by  experience, 
are  the  necessary  and  immediate  consequences,  being, 
indeed,  nothing  more  than  partial  and  complementary 
statements  of  it.  The  limitation  of  gaseous  volume  be- 
ing produced  by  pressure  alone — the  cohibition  of  the 
bulk  of  a  gas  being  due  solely  to  pressure — it  follows 
that  it  must  be  proportional  to  it ;  in  other  words,  that 
the  volume  of  a  gas  must  be  inversely  as  the  pressure ; 
and  this  is  the  law  of  Boyle  or  Mariotte.  Again  :  tem- 
perature is  measured  by  the  uniform  expansion  of  a 
column  of  gas  (in  the  air-thermometer) ;  hence,  if  all 
gases  expand  equally,  temperature  is.  proportional  to  the 
volume  of  a  gas  and  conversely;  this  is  the  law  of 
Charles.* 

*  One  of  the  strangest  incidents  in  the  history  of  physics  is  the  grave 
discussion  of  the  question  respecting  the  true  law  of  gaseous  expansion. 
"According  to  Gay-Lussac,"  says  Balfour  Stewart  (Treatise  on  Heat, 
p.  60),  "  the  augmentation  of  volume  which  a  gas  receives  when  the  tem- 
perature increases  1°  is  a  certain  fixed  proportion  of  its  initial  volume  at 
0°  C.  ;  while,  according  to  Dalton,  a  gas  at  any  temperature  increases  in 
volume  for  a  rise  of  1°  by  a  constant  fraction  of  its  volume  at  that  tem- 
perature. .  .  .  The  dilatation  of  gases  has  since  been  investigated  by 
Rudberg,  Dulong  and  Petit,  Magnus  and  Regnault,  and  the  result  of  their 
labors  leaves  little  doubt  that  Gay-Lussac's  method  of  expressing  the  law 
is  much  nearer  the  truth  than  Dalton's."  Inasmuch  as  the  experiments^ 
6 


118  CONCEPTS  OF  MODERN  PHYSICS. 

The  foregoing  real  definition  (i.  e.,  exhibition  of  the 
properties)  of  a  gas  applies  only  to  ideal  or  perfect 
gases.  In  actual  experience  we  meet  with  no  gas 
which,  in  the  absence  of  pressure,  expands  with  abso- 
lute uniformity ;  and  for  that  reason  we  do  not  know 
experimentally  of  any  gas  behaving  in  strict  conformity 
to  the  laws  of  Boyle  and  Charles.  Moreover,  we  are 
unable  directly  to  observe  a  gas  which  is  wholly  free 
from  pressure ;  the  datum  of  experience  is  simply  that 
gases  expand  (other  things  being  equal)  in  proportion 
to  the  diminution  of  the  pressure  to  which  they  are 
subjected.  But  in  the  case  of  many  gases— those  which 

of  Rudberg  and  others  were  necessarily  made  on  the  supposition  that  the 
coefficient  of  expansion  was  the  same  for  all  gases  (the  question  relating, 
not  to  the  expansion  of  some  particular  gas,  but  of  gases  generally),  and, 
as  the  standard  temperatures  were  those  of  the  air-thermometer,  it  would 
have  been  surprising,  indeed,  if  the  result  had  been  confirmatory  of  Dai- 
ton's  view.  A  thermometer  is  graduated  by  dividing  a  given  length  of  a 
tube  of  even  bore  into  equal  parts.  It  is  clear,  therefore,  that  the  incre- 
ment of  volume  resulting  from  the  expansion  of  the  air  in  such  a  tube 
through  one  degree  is  a  fixed  part  of  a  constant  volume  initially  assumed, 
and  not  of  a  constantly  increasing  volume ;  and  the  same  thing  is,  of 
course,  true  of  any  other  gas  if  it  expands  at  the  same  rate.  Dalton's 
form  of  the  law  of  expansion  would  yield  the  following  remarkable  series 
of  equal  ratios — in  which  the  first  represents  the  rate  of  expansion  of  air 
in  the  thermometer,  and  the  others  stand  for  the  rate  (or  rather  rates)  of 
expansion  of  the  gas  under  examination  (a  being  the  linear  expansion  of 
the  air  in  the  thermometer,  v  its  initial  volume,  a'  the  corresponding  ex- 
pansion in  the  gas  under  examination,  v'  its  initial  volume) :  -  =  —  = 

i  i  ii  V         V 

ct  d  ,  a  ci 

-. -,  =  - — — ;  =  —. —  =  —, — ,  etc.,  etc.    The  attempts  at  an 

v'  +  a       v  +  2a       v  +  3a      »'  +  4a' 

experimental  solution  of  the  question  here  referred  to  are  suggestive,  by 
the  way,  of  a  doubt  as  to  the  correctness  of  the  prevailing  systems  of  ther- 
mometry,  which  are  founded  on  the  assumption  of  equalities  of  volume- 
ratios  in  which  one  of  the  terms  is  constant  while  the  other  is  variable, 
i.  e.,  of  fractions  which  have  the  same  numerator,  but  different  denomina- 
tors. These  suggestions  are  but  imperfectly  met  by  the  reflection  that 
the  bores  of  our  thermometrical  tubes  are  very  small. 


THE  KINETIC  THEORY  OF  GASES. 

are  either  wholly  incoercible,  or  coercible  (i.  e.,  reduci- 
ble to  the  liquid  or  solid  state)  with  great  difficulty,  and 
of  nearly  all  gases  at  very  high  temperatures — the  devi- 
ation from  uniformity  of  expansion  is  very  slight. 

Now,  how  does  the  kinetic  theory  of  gases  explain 
the  experiential  fact  or  facts  just  stated  ?  It  professes 
to  explain  them  on  the  basis  of  at  least  three  arbitrary 
assumptions,  not  one  of  which  is  a  datum  of  experience, 
viz. : 

1.  That  a  gas  is  composed  of  solid  particles  which 
are  indestructible  and  of  constant  mass  and  volume. 

2.  That  these  constituent  particles   are  absolutely 
elastic. 

3.  That  these  particles  are  in  perpetual  motion,  and, 
except  at  very  small  distances,  in  no  wise  act  upon  each 
other,  so  that  their  motions  are   absolutely  free  and 
therefore  rectilinear. 

I  refrain  from  adding  a  fourth  assumption — that  of 
the  absolute  equality  of  the  particles,  in  mass  at  least — 
because  it  is  claimed  (though  unjustifiably)  to  be  a  cor- 
ollary from  the  other  assumptions. 

The  first  of  these  assumptions  has  been  sufficiently 
considered  in  the  last  chapter.  The  second  assumption 
asserts  the  absolute  elasticity  of  the  constituent  solid 
particles.  What  is  the  import  and  scope  of  this  assump- 
tion ?  The  elasticity  of  a  solid  body  is  that  property  by 
means  of  which  it  occupies,  and  tends  to  occupy,  por- 
tions of  space  of  determinate  volume  and  figure,  and 
therefore  reacts  against  any  force  or  stress  producing, 
or  tending  to  produce,  an  alteration  of  such  volume  or 
figure  with  a  counter-force  or  stress  which,  in  the  case 
of  perfect  elasticity,  is  exactly  proportional  to  the  alter- 
ing force.  Now,  it  is  seen  at  once  that  the  property — 
the  fact — thus  assumed  in  the  constituent  solid  includes 


120  CONCEPTS  OF  MODERN  PaYSICS. 

the  very  fact  to  be  accounted  for  in  the  gas.  A  perfect 
gas  reacts  against  a  stress  tending  to  reduce  its  volume 
with  a  spring  proportional  to  the  stress ;  and  for  this 
reason  gases  are  denned  as  elastic  fluids.  This  resili- 
ence of  the  gas  against  diminution  of  volume  is  obvi- 
ously a  simpler  fact  than  the  rebound  of  a  solid  against 
both  diminution  and  increase  of  volume,  in  addition  to 
the  reaction  against  a  change  of  figure.  The  resistance 
to  several  kinds  of  change  implies  a  greater  number  of 
forces,  and  is  therefore  a  more  complex  phenomenon, 
than  the  resistance  to  one  kind  of  change.* 

It  thus  appears  that  the  presupposition  of  absolute 
elasticity  in  the  solids,  whose  aggregate  is  said  to  con- 
stitute a  gas,  is  a  flagrant  violation  of  the  first  condition 
of  the  validity  of  an  hypothesis — the  condition  which 
requires  a  reduction  of  the  number  of  unrelated  ele- 
ments in  the  fact  to  be  explained,  and  therefore  forbids 
a  mere  reproduction  of  this  fact  in  the  form  of  an  as- 
sumption, and  a  fortiori  a  substitution  of  several  arbi- 
trary assumptions  for  one  fact.  Manifestly  the  expla- 
nation offered  by  the  kinetic  hypothesis,  in  so  far  as  its 
second  assumption  lands  us  in  the  very  phenomenon 
from  which  it  starts,  the  phenomenon  of  resilience,  is 
(like  the  explanation  of  impenetrability,  or  of  the  com- 

*  It  may  be  said  that  the  greater  simplicity  of  the  properties  of  a  gas 
is  purely  conceptual.  The  identification  of  concepts  with  facts  is  un- 
doubtedly the  great  fundamental  error  of  speculation ;  but  we  are  now 
dealing  with  the  conceptual  elements  of  the  hypothesis  under  discussion. 
The  opinion  that  a  solid  of  constant  volume  (or,  more  accurately  ex- 
pressed, of  variable  volume,  expanding  or  contracting  to  a  fixed  volume 
proprio  motu)  is  a  simpler  thing  than  a  uniformly  expanding  body  is 
certainly  not  based  upon  any  fact  of  experience,  but  is  a  mere  prejudice 
of  the  intellect  akin  to  the  notion  that  a  body  at  rest  is  a  simpler  phenom- 
enon than  a  body  in  uniform  motion,  and  generally  that  rest  is  simpler 
than  motion.  This  prejudice  has  its  root  in  our  habitual  oblivion  of  the 
essential  relativity  of  all  phenomena,  which  will  be  discussed  hereafter. 


THE  KINETIC  THEORY  OF  GASES. 

bination  of  elements  in  definite  proportions  by  the 
atomic  theory)  simply  the  illustration  of  idem  per  idem, 
and  the  very  reverse  of  a  scientific  procedure.  It  is  a 
mere  versatio  in  loco — movement  without  progress.  It 
is  utterly  vain ;  or  rather,  inasmuch  as  it  complicates 
the  phenomenon  which  it  professes  to  explicate,  it  is 
worse  than  vain — a  complete  inversion  of  the  order  of 
intelligence,  a  resolution  of  identity  into  difference,  a 
dispersion  of  the  One  into  the  Many,  an  unraveling  of 
the  Simple  into  the  Complex,  an  interpretation  of  the 
Known  in  terms  of  the  Unknown,  an.  elucidation  of 
the  Evident  by  the  Mysterious,  a  reduction  of  an  os- 
tensible and  real  fact  to  a  baseless  and  shadowy  phan- 
tom.* 

Waiving  the  question  already  discussed,  whether  or 
not  the  assumed  absolute  solidity  and  constancy  of  vol- 
ume of  the  supposed  constituent  particles  are  consistent 
(in  the  light  of  the  mechanical  theory 'generally)  with 
their  absolute  elasticity,  I  proceed  to  consider  the  third 
assumption  of  the  kinetic  hypothesis.  This  assumption 
is  an  unavoidable  supplement  to  the  initial  theoretical 

*  All  theorists  who  attempt  to  account  for  a  physical  fact  by  a  multi- 
plication of  arbitrary  assumptions  in  which  the  fact  itself  is  reproduced 
are  liable  to  Aristotle's  acute  animadversion  upon  the  Platonic  doctrine 
of  ideas — their  endeavors  are  as  nugatory  as  those  of  a  person  who,  for 
the  purpose  of  facilitating  the  operation  of  counting,  begins  by  multiply- 
ing his  numbers — of  8e  ras  tSeas  curias  Tt&e/uej/ot  irpurov  /j.ev 
To>v8l  rS>v  ovTtav  AajSeTi/  ras  curias  erepa  Todrois  fcra  rov  apt3/toi/ 
&(rirep  si  ns  api^/j.TjO'ai.  fiov\6jj.evos  s\arr6vwv  /j.ev  OVTWV  O'IOITO  ^ 
&ai  ir\et(i>  5e  iroi-f)<ras  ap&polri.  Met.,  A.  9,  990,  et  seq.  Occam's  rule 
"  Entia  non  sunt  muUiplicanda  praeter  neccssitatem  "  has  its  applications 
in  physics  no  less  than  in  metaphysics ;  and  there  are  physical  theories 
of  which  Michel  Montaigne,  if  he  lived  to-day,  would  say  what  he  said  of 
certain  scholastic  vagaries,  three  hundred  years  ago :  "  On  esehanye  un 
mot  pour  un  aultre  mot,  et  souvent  plus  incogneu.  .  .  .  Pour  satisfaire  d 
un  doubte,  Us  rrfen  donncnt  trois  ;  Jest  la  teste  d>  Hydra.  .  .  .  Nous  com- 
muniquons  une  question  ;  on  nous  en  redonne  une  ruchee."  Essais,  iii,  1 3. 


122  CONCEPTS  OF  MODERN  PHYSICS. 

complication  of  the  phenomenon  of  elasticity,  produced 
by  the  arbitrary  substitution  of  the  resilience  of  a  solid 
against  increase  or  diminution  of  volume  and  change 
of  figure  for  the  reaction  of  a  gas  against  diminution 
of  volume  alone.  To  get  rid  of  one  gratuitous  feature 
of  the  hypothesis  (the  addition  of  the  rebound  against 
dilatation  and  distortion  to  that  against  compression) 
and  to  bring  it  into  conformity  with  the  fact  to  be  ex- 
plained, it  becomes  necessary  to  add  another  arbitrary 
feature — to  endow  the  parts  with  incessant  rectilinear 
motion  in  all  directions.  In  respect  to  this  assumption, 
which,  like  other  assumptions  of  the  mechanical  theory, 
is  based  upon  a  total  disregard  of  the  relativity  and 
consequent  mutual  dependence  of  natural  phenomena, 
it  is  to  be  said,  for  the  present,  that  it  is  utterly  gratui- 
tous, and  not  only  wholly  unwarranted  by  experience, 
but  out  of  all  analogy  with  it.  Bodies  which,  except 
on  the  very  verge  of  immediate  contact,  move  inde- 
pendently without  mutual  attraction  or  repulsion  or  any 
sort  of  mutual  action  and  thus  present  perfect  realiza- 
tions of  the  abstract  concept  of  free  and  ceaseless  recti- 
linear motion,  are  unheard-of  strangers  in  the  wide  do- 
main of  sensible  experience.  So  complete  an  abandon- 
ment of  the  analogies  of  experience  is  all  the  more 
surprising  in  view  of  the  circumstance  that  the  atomic 
hypothesis,  whereof  the  kinetic  theory  of  gases  is  a 
branch,  is  confessedly  a  concretion  of  suggestions  de- 
rived from  celestial  mechanics.  There  is  hardly  a  trea- 
tise on  modern  physics  in  which  the  atoms  or  molecules 
are  not  compared  to  planetary  or  stellar  systems.  "  A 
.  compound  atom,"  says  Jevons,*  "  may  perhaps  be  com- 

*  Principles  of  Science,  i,  453.  In  Arwed  Walter's  Untersuchungen 
ueber  Molecularmechanik,  p.  216,  the  system  of  Jupiter  and  his  satel- 
lites is  called  a  "  planetary  molecule." 


THE  KINETIC  THEORY  OF  GASES.  123 

pared  with  a  stellar  system,  each  star  a  minor  system  in 
itself."  But  the  bodies  with  which  celestial  mechanics 
deal  are  all  subject  to  the  law  of  attraction ;  and  the 
import  of  the  very  first  theorem  of  Newton's  Prin- 
cipia  is,  that  these  bodies,  if  their  motions  are  at  any 
moment  out  of  the  same  straight  line,  can  never  collide, 
but  must  always  move  in  curved  orbits  at  a  distance 
from  each  other.  Oblique  impacts  between  them  pro- 
ductive of  rotations  as  well  as  of  deviations  from  their 
paths  before  impact,  as  they  are  imagined  by  Clausius 
and  the  other  promoters  of  the  kinetic  theory,  are  im- 
possible. And  this  is  true,  not  only  when  the  mutual 
actions  of  the  bodies  vary  inversely  as  the  squares  of 
their  distances,  but  whenever  they  vary  as  any  higher 
power  of  these  distances — &  proposition  to  be  borne  in 
mind  in  view  of  certain  speculations  of  Boltzrnann, 
Stefan,  and  Maxwell,  of  which  I  shall  presently  speak. 

There  is  another  very  extraordinary  and,  in  the  light 
of  all  the  teachings  of  science,  unwarrantable  feature  in 
the  assumption  respecting  the  movements  of  the  alleged 
solid  constituent  particles.  I  allude  to  the  absolute  dis- 
continuity between  the  violent  mutual  action  attributed 
to  these  particles  during  the  few  instants  of  time  be- 
fore and  after  their  collisions,  and  their  total  freedom 
from  mutual  action  during  the  comparatively  long 
periods  of  their  rectilinear  motion  along  "  free  paths." 
And  this  leads  me  to  say  a  few  words  in  regard  to 
certain  subsidiary  assumptions  made  by  Maxwell  and 
others  in  order  to  account  for  the  anomalies  exhibited 
by  gases  of  different  degrees  of  coercibility  in  their  de- 
viations from  Boyle's  and  Charles's  law.  Maxwell  as- 
sumes that  the  gas-molecules  are  neither  strictly  spheri- 
cal nor  absolutely  elastic,  and  that  their  centers  repel 
each  other  with  a  force  inversely  proportional  to  the 


124  CONCEPTS  OF  MODERN  PHYSICS. 

fifth  power  of  their  distance;*  while  Stefanf  endeav- 
ors to  adjust  the  hypothesis  to  the  phenomena  in  ques- 
tion by  postulating  that  the  molecules  are  absolutely 
elastic  and  perfect  spheres  whose  diameters  are  inverse- 
ly proportional  to  the  fourth  roots  of  the  absolute  tem- 
peratures of  the  gases.  These  assumptions,  which  are 
fatal  to  all  claims  of  simplicity  preferred  on  behalf  of 
the  kinetic  hypothesis,  are  in  no  sense  an  outgrowth  of 
its  original  postulates;  both  are  purely  gratuitous  as 
well  as  without  experiential  analogy,  and  the  first  of 
them,  that  of  Maxwell,  is  in  direct  defiance  of  all  the 
inductions  from  the  wide  range  of  actual  observation. 
They  are  both  mere  stop-gaps  of  the  hypothesis,  peace- 
offerings  for  its  non-congruence  with  the  facts,  pure 
inventions  to  satisfy  the  emergencies  created  by  the 
hypothesis  itself. 

It  were  work  of  supererogation  to  review  in  detail 
the  logical  and  mathematical  methods  by  which  it  is 
attempted,  from  an  hypothesis  resting  on  such  founda- 
tions, to  deduce  formulae  corresponding  to  the  facts 
of  experience.  I  may  be  permitted  to  say,  however, 
that  the  methods  of  deduction  are  only  less  extraordi- 
nary than  the  premisses.  To  account  for  the  laws  of 
Boyle  and  Charles  resort  is  had  to  the  calculus  of  prob- 
abilities, or,  as  Maxwell  terms  it,^:  the  method  of  sta- 
tistics. It  is  alleged  that,  although  the  individual  mol- 
ecules move  with  unequal  velocities,  either  because 
these  velocities  were  originally  unequal,  or  because  they 

*  Since  this  was  written,  Maxwell  himself  has  abandoned  this  assump- 
tion as  not  conformable  to  the  facts. 

•f  Ueber  die  dynamische  Diffusion  der  Gase.     Sitzungsberichte  der 
kaiserlichen  Akademie  der  Wissenschaften,  Mathem.  naturw.  Classe,  vol. 
Ixv,  p.  323.     Cf.  also  Boltzmann,  Ueber  das  Wirkungsgesetz  der  Moleku- 
larkraefte,  Sitzungsberichte,  etc.,  vol.  Ixvi,  p.  213. 
\  Theory  of  Heat,  p.  288. 


THE  KINETIC  THEORY  OF  GASES. 

have  become  unequal  in  consequence  of  the  encounters 
between  them,  nevertheless,  there  will  be  an  average  of 
all  the  velocities  belonging  to  the  molecules  of  a  system 
(i.  e.,  of  a  gaseous  body)  which  Maxwell  calls  the  "  ve- 
locity of  mean  square."  The  pressure,  on  this  supposi- 
tion, is  proportional  to  a  product  of  the  square  of  this 
average  velocity  into  the  number  of  molecules  multi- 
plied by  the  mass  of  each  molecule.  The  product  of 
the  number  of  molecules  into  the  mass  of  each  mole- 
cule is  then  replaced  by  the  density — in  other  words, 
the  whole  molecular  assumption  is,  for  the  nonce, 
abandoned — and  the  velocity  is  eliminated  as  represent- 
ing the  temperature;  it  follows,  of  course,  that  the 
pressure  is  proportional  to  the  density. 

Similar  procedures  lead  to  the  law  of  Charles  and 
the  "  law  "  of  Avogadro  (according  to  which  the  num- 
ber of  molecules  in  any  two  equal  volumes  of  gases  of 
whatever  kind  is  the  same  at  the  same  temperatures 
and  pressures — a  law  which  is  itself  a  mere  hypothesis). 
It  is  claimed,  on  statistical  grounds  again,  that  not  only 
the  average  velocity  of  a  number  of  molecules  in  a 
given  gaseous  body  is  the  same,  but  that  "  if  two  sets 
of  molecules,  whose  mass  is  different,  are  in  motion  in 
the  same  vessel,  they  will,  by  their  encounters,  ex- 
change energy  with  each  other  till  the  average  kinetic 
energy  of  a  single  molecule  of  either  set  is  the  same."  * 
"  This,"  says  Maxwell,  "  follows  from  the  same  inves- 
tigation which  determines  the  law  of  distribution  of 
velocities  in  a  single  set  of  molecules."  All  this  being 
granted,  the  law  of  Charles  and  the  law  of  Avogadro 
(called  by  Maxwell  the  law  of  Gay-Lussac)  are  readily 
derived.  And  at  the  end  of  these  devious  courses  of 
deduction  Maxwell  adds  a  disquisition  on  the  properties 

*  Maxwell,  /.  c.,  p.  289  seq. 


126  CONCEPTS  OF  MODERN  PHYSICS. 

of  molecules,  in  which  he  claims  to  have  made  it  evi- 
dent that  the  molecules  of  the  same  substance  are  "  un- 
alterable by  the  processes  which  go  on  in  the  present 
state  of  things,  and  every  individual  of  the  same  species 
is  of  exactly  the  same  magnitude  as  though  they  had 
all  been  cast  in  the  same  mold,  like  bullets,  and  not 
merely  selected  and  grouped  according  to  their  size, 
like  small  shot,"  and  that,  therefore,  as  he  expresses  it 
in  another  place,*  they  are  not  the  products  of  any  sort 
of  evolution,  but,  in  the  language  of  Sir  John  Herschel, 
"  have  the  essential  character  of  manufactured  articles." 

]$Tow,  on  what  logical,  mathematical,  or  other  grounds 
is  the  statistical  method  applied  to  the  velocities  of  the 
molecules  in  preference  to  their  weights  and  volumes  ? 
What  reason  is  given,  or  can  be  given,  why  the  masses 
of  the  molecules  should  not  be  subjected  to  the  process 
of  averaging  as  well  as  their  motions?  None  what- 
ever. And,  in  the  absence  of  such  reason,  the  deduc- 
tions of  the  kinetic  theory,  besides  being  founded  on 
rickety  premisses,  are  delusive  paralogisms. 

Upon  these  considerations  I  do  not  hesitate  to  de- 
clare that  the  kinetic  hypothesis  has  none  of  the  charac- 
teristics of  a  legitimate  physical  theory.  Its  premisses 
are  as  inadmissible  as  the  reasoning  upon  them  is  in- 
conclusive. It  postulates  what  it  professes  to  explain  ; 
it  is  a  solution  in  terms  more  mysterious  than  the  prob- 
lem— a  solution  of  an  equation  by  imaginary  roots  of 
unknown  quantities.  It  is  a  pretended  explanation,  of 
which  it  were  unmerited  praise  to  say  that  it  leaves  the 
facts  where  it  found  them,  and  is  obnoxious  to  the  old 
Horatian  stricture  :  nil  agit  exemplum,  litem  quod  lite 
resolvit. 

*  Bradford  Lecture  on  the  Theory  of  Molecules,  cf .  Popular  Science 
Monthly,  January,  1874. 


THE  KINETIC  THEORY  OF  GASES.  '127 

Much  is  said  about  the  support  derived  by  the 
kinetic  theory  of  gases  from  the  revelations  of  the 
spectroscope.  The  spectra  of  gases,  unlike  those  of 
solids  and  liquids,  are  not  continuous,  but  consist  of 
distinct  colored  lines  or  bands — showing,  as  is  claimed, 
that  in  gases  the  vibrations  of  molecules  do  not  inter- 
fere; that  incandescent  gases  emit  distinct  kinds  of 
light  and  not  (as  Jevons  expresses  it)  luminar  noises, 
because  there  is  no  clashing  of  the  molecules  disturbing 
the  natural  periods  of  vibration.*  The  spectroscope  is, 
no  doubt,  the  most  important  witness  yet  called  on  be- 
half of  the  kinetic  theory ;  but  the  testimony  of  this 
witness  is  not  all  in  its  favor.  "  The  spectroscope," 
says  Maxwell  himself,  f  "shows  that  some  molecules 
can  execute  a  great  many  different  kinds  of  vibrations. 
They  must,  therefore,  be  systems  of  very  considerable 
complexity,  having  far  more  than  six  variables.  Now, 
every  additional  variable  introduces  an  additional 
amount  of  capacity  for  internal  motion  without  in- 
creasing the  external  pressure.  Every  additional  vari- 
able, therefore,  increases  the  specific  heat,  whether 
reckoned  at  constant  pressure  or  constant  volume.  So 
does  any  capacity  which  the  molecule  may  have  for 
storing  up  energy  in  the  potential  form.  But  the  cal- 
culated specific  heat  is  already  too  great  when  we  sup- 
pose the  molecule  to  consist  of  two  atoms  only.  Hence 

*  According  to  the  latest  interpretation  of  spectroscopic  phenomena, 
the  continuity  or  discontinuity  of  a  spectrum  is  indicative,  not  so  much  of 
the  state  of  aggregation,  as  of  the  molecular  complexity  of  the  body  ex- 
amined. It  is  said  that  a  body  yields  a  spectrum  of  lines  when  its  mole- 
cules contain  but  a  few  atoms  each ;  that,  when  they  contain  more,  the 
spectrum  presents  the  appearance  of  fluted  bands  ;  and  that  the  spectrum 
is  continuous  when  each  molecule  comprises  a  great  number  of  atoms. 

f  On  the  Dynamical  Evidence  of  the  Molecular  Constitution  of  Bod- 
ies, Nature,  March  4  and  11,  1876,  Nos.  279,  280. 


128  CONCEPTS  OF  MODERN  PHYSICS. 

every  additional  degree  of  complexity  which  we  attrib- 
ute to  the  molecule  can  only  increase  the  difficulty  of 
reconciling  the  observed  with  the  calculated  value  of 
the  specific  heat." 

It  may  seem  strange  that  so  many  of  the  leaders  of 
scientific  research,  who  have  been  trained  in  the  severe 
schools  of  exact  thought  and  -rigorous  analysis,  should 
have  wasted  their  efforts  upon  a  theory  so  manifestly 
repugnant  to  all  scientific  sobriety — an  hypothesis  in 
which  the  very  thing  to  be  explained  is  but  a  small 
part  of  its  explanatory  assumptions.  But  even  the 
intellects  of  men  of  science  are  haunted  by  pre-scientific 
survivals,  not  the  least  of  which  is  the  inveterate  fancy 
that  the  mystery  by  which  a  fact  is  surrounded  may  be 
got  rid  of  by  minimizing  the  fact  and  banishing  it  to 
the  regions  of  the  Extra-sensible.  The  delusion  that 
the  elasticity  of  a  solid  atom  is  in  less  need  of  explana- 
tion than  that  of  a  bulky  gaseous  body  is  closely  related 
to  the  conceit  that  the  chasm  between  the  world  of 
matter  and  that  of  mind  may  be  narrowed,  if  not 
bridged,  by  a  rarefaction  of  matter,  or  by  its  resolu- 
tion into  "  forces."  The  scientific  literature  of  the  day 
teems  with  theories  in  the  nature  of  attempts  to  con- 
's vert  facts  into  ideas  by  a  process  of  dwindling  or  sub- 
tilization.  All  such  attempts  are  nugatory ;  the  intan- 
gible specter  proves  more  troublesome  in  the  end  than 
the  tangible  presence.  Faith  in  spooks  (with  due  re- 
spect be  it  said  for  Maxwell's  thermo-dynamical  "  de- 
mons" and  for  the  population  of  the  "Unseen  Uni- 
verse ")  is  unwisdom  in  physics  no  less  than  in  pneu- 
matology. 


CHAPTEE  IX. 

THE    RELATION     OF    THOUGHTS    TO     THINGS. THE    FORMA- 
TION   OF    CONCEPTS. METAPHYSICAL    THEORIES. 

IT  has  become  evident,  I  take  it,  in  the  course  of 
the  preceding  discussions,  that,  while  modern  physical 
science  is  professedly  an  endeavor  to  reduce  the  phe- 
nomena of  nature  to  the  elements  of  mass  and  motion, 
and  thus  to  exhibit  them  as  results  or  phases  of  me- 
chanical action — claiming,  on  this  ground,  to  be  the- 
only  mode  of  dealing  with  these  phenomena  that  is  not 
in  its  nature  metaphysical — nevertheless  all  the  depart- 
ments of  science  which  have  made  decided  advances 
beyond  the  first  classificatory  stage  proceed  upon  as- 
sumptions and  lead  to  consequences  inconsistent  with 
the  object  of  this  endeavor  and  with  the  fundamental 
principles  of  the  mechanical  theory.  We  find  ourselves 
in  the  midst  of  a  confusion,  therefore,  which  is  to  be 
cleared  up,  if  at  all,  by  an  inquiry  into  the  origin  of 
this  theory  and  by  a  determination  of  its  attitude  tow- 
ard the  laws  of  thought  and  the  forms  and  conditions 
of  its  evolution. 

The  account  given,  by  ordinary  psychologists  and 
logicians,  of  the  nature  and  operations  of  thought  may, 
so  far  as  it  bears  upon  the  matter  now  under  considera- 
tion, be  compressed  into  a  few  sentences.  Thought,  in 
its  most  comprehensive  sense,  is  the  establishment  or 
recognition  of  relations  between  phenomena.  Foremost 


130  CONCEPTS  OF  MODERN  PHYSICS. 

among  these  relations — the  foundation,  in  fact,  of  all 
others,  such  as  those  of  exclusion  and  inclusion,  coexist- 
ence and  sequence,  cause  and  effect,  means  and  end — 
are  the  relations  of  identity  and  difference.  The  differ- 
ence between  phenomena  is  a  primary  datum  of  sensa- 
tion. The  very  act  of  sensation  is  based  upon  it.  It  is 
one  of  the  many  acute  observations  of  Hobbes  that  "  it 
is  all  one  to  be  always  sensible  of  the  same  thing  and 
not  to  be  sensible  of  anything."*  "  We  only  know  any- 
thing," says  J.  S.  Mill,f  "  by  knowing  it  as  distin- 
guished from  something  else ;  all  consciousness  is  of 
difference;  two  objects  are  the  smallest  number  re- 
quired to  constitute  consciousness ;  a  thing  is  only  seen 
to  be  what  it  is  by  contrast  with  what  it  is  not." 

"While  the  apprehension  of  phenomenal  difference 
(which,  however,  may  be,  and  in  most  cases  is,  replaced 
by  its  reproduction  in  memory)  is  the  basis  or  pre- 
requisite of  thought,  thought  proper,  i.  e.,  discursive 
thought,  begins  with  the  apprehension  of  identity  amid 
phenomenal  difference.  Objects  are  perceived  as  differ- 
ent ;  they  are  conceived  as  identical  by  an  attention  of 
the  mind  to  their  point  or  points  of  agreement.  They 
are  thus  classified,  the  points  of  agreement,  i.  e.,  those 
properties  of  the  objects  of  cognition  which  belong  to 
them  in  common,  serving  as  the  basis  of  classification. 
When  the  number  of  objects  classified  is  great,  and 
some  of  these  objects  have  more  properties  in  common 
than  others,  a  series  of  classes  is  formed.  The  objects 
are  first  divided  into  groups  (called  by  the  logicians  in- 
fimce  species)  severally  embracing  such  objects  as  are 
characterized  by  the  greatest  number  of  common  prop- 

*  "  Sentire  semper  idem  et  non  sentire  ad  idem  recidunt."    Hobbes, 
Physica,  iv,  25  (opp.,  ed.  Molesworth,  vol.  i,  p.  321. 

\  Examination  of  Sir  William  Hamilton's  Phil.  (Am.  ed.),  vol.  i,  p.  14. 


RELATION  OF  THOUGHTS  TO  THINGS.  '131 

erties  consistent  with  their  difference ;  these  groups 
are  then  collected  and  distributed  into  higher  groups  or 
species  having  a  less  number  of  properties  in  common, 
and  so  on,  until  we  arrive  at  the  least  number  of  prop- 
erties in  which  all  the  objects  embraced  in  (logice  sub- 
sumed under)  the  infimce  species  and  the  intermediate 
species  agree,  so  as  to  characterize  the  highest  class,  or 
summum  genus. 

From  this  it  follows  that,  in  proportion  as  we  ascend 
the  scale  of  classification  from  the  infimce  species  to  the 
summum  genus,  the  number  of  objects  embraced  in  the 
successive  classes  (species  or  genera)  increases,  while 
the  number  of  characteristic  properties  decreases.  Now, 
the  complement  of  properties  characteristic  of  a  par- 
ticular class  is  termed  a  concept;  the  number  of  objects 
denoted  by  each  concept  is  called  its  extension  or 
breadth;  and  the  number  of  properties  (which,  as  con- 
stituents of  a  concept,  bear  the  name  of  attributes) 
connoted  by  it  its  extension,  comprehension  or  depth ; 
whence  springs  the  law  of  logic  that,  the  greater  the 
extension  of  a  concept,  i.  e.,  the  greater  the  number  of 
objects  denoted,  the  less  its  comprehension,  i.  e.,  the 
number  of  attributes  connoted ;  or,  expressed  with 
mathematical  accuracy,  that  the  extension  varies  in  ge- 
ometrical ratio  inversely  as  the  comprehension  varies  in 
arithmetical  ratio.* 

It  is  readily  seen  that  the  ascent  from  a  lower  (more 
comprehensive,  but  less  extensive)  to  a  higher  (more  ex- 
tensive but  less  comprehensive)  class  is  effected  by  a 
progressive  segregation  and  ideal  union  of  those  attri- 
butes which  the  respective  classes  have  in  common  ;  and 
this  process  is  termed  abstraction. 

*  For  an  exact  statement  of  the  law  in  question,  see  Drobisch,  Neue 
Darstellung  der  Logik,  Logisch-mathematischer  Anhang  (third  ed.,  p.  206). 


132  CONCEPTS  OF  MODERN   PHYSICS. 

In  the  sense  of  the  foregoing  exposition,  thought 
proper  has  been  defined  as  uthe  act  of  knowing  or 
judging  of  things  by  means  of  concepts,"  *  a  concept 
being  "  a  collection  of  attributes  united  by  a  sign  and 
representing  a  possible  object  of  intuition."!  This  defi- 
nition of  a  concept,  however,  is  obnoxious  to  criticism, 
as  being  either  too  wide  or  too  narrow.  It  may  be  said, 
on  the  one  hand,  to  be  too  wide :  for  it  applies  to  the 
total  array  of  attributes  constituting  the  mental  repre- 
sentation of  a  single  object,  without  reference  to  the 
question  whether  or  not  they  are  shared  by  any  other 
object,  as  well  as  to  the  factitious  selection  or  collection 
of  attributes  characteristic  of  a  class,  i.  e.,  of  a  plurality 
of  objects.  In  other  words,  it  is  a  definition  of  singular 
concepts  (expressed  by  singular  terms)  as  well  as  of  gen- 
eral concepts  (expressed  by  general  terms,  or,  as  Mill 
would  say,  class  names).  In  the  language  of  the  old 
logicians,  it  includes  infimce  species,  and  may  stand  for 
any  singular  object  or  singular  quality,  irrespective  of 
the  fact  or  degree  of  its  ge&isality.  This  criticism 
would  be  avoided  by  defining  a  concept,  with  Sir  Wil- 
liam Hamilton,;);  as  "  the  cognition  of  the  general  char- 
acter, point  or  points  in  which  a  plurality  of  objects 
coincide."  On  the  other  hand,  the  word  "  concept "  is 
very  generally  employed  in  a  sense  for  which  Hansel's 
definition  is  too  narrow.  German  logicians,  for  example, 
habitually  designate  not  only  every  mental  reproduc- 
tion of  a  presentation  of  sense,  in  so  far  as  it  is  or  may 
be  an  element  of  a  judgment  or  logical  proposition,  as 
a  concept  (^Begriff\  but  also  the  last  result  of  any  series 
of  abstractions.  And  the  last  results  of  abstraction,  the 

*  Mansel,  Prolegomena  Logica,  p.  22. 

f  /&.,  p.  60. 

\  Lectures  on  Logic,  p.  87. 


RELATION  OF  THOUGHTS  TO  THINGS. 

summa  genera,  are  excluded  by  the  definition  of  Man- 
sel.  It  is  neither  necessary  nor  practicable  here  to  at- 
tempt a  minute  discussion  of  the  questions  arising  upon 
these  divergences  in  the  use  of  terms ;  nor  can  I  stop  to 
weigh  the  objections  recently  urged  by  Tauschinsky, 
Lotze,  Sigwart,  Wundt  and  others  to  the  theory  of  con- 
ception as  founded  upon  classification  or  subsumption. 
The  controversies  on  this  head  between  the  logicians  of 
the  old  and  those  of  the  new  school,  as  well  as  the  in- 
terminable disputes  between  the  nominalists  and  the 
conceptualists  to  which  so  large  a  space  is  devoted  in 
the  writings  of  J.  S.  Mill,*  are  in  the  main  mere  wars 
of  words,  and  the  points  of  disagreement  are  foreign  to 
the  investigation  upon  which  I  am  about  to  enter.  To 
one  or  two  of  these  points  I  may  have  occasion  to  recur 
hereafter;  for  the  present  my  brief  summary  of  the 
incidents  of  logical  conception  is  to  serve  only  as  a  clew 
to  the  meaning  of  certain  logical  terms  I  am  con- 
strained to  employ,  whenever  this  meaning  is  not  suffi- 
ciently apparent  from  the  context. 

Now,  in  any  discussion  of  the  operations  of  thought, 
it  is  of  the  utmost  importance  to  bear  in  mind  the  fol- 
lowing irrefragable  trutts,  some  of  which — although  all 
of  them  seem  to  be  obvious — have  not  been  clearly  ap- 
prehended until  very  recent  times : 

1.  Thought  deals,  not  with  things  as  they  are,  or 
are  supposed  to  be,  in  themselves,  but  with  our  mental 
representations  of  them.  Its  elements  are,  not  pure 
objects,  but  their  intellectual  counterparts.  What  is 
present  in  the  mind  in  the  act  of  thought  is  never  a 
thing,  but  always  a  state  or  states  of  consciousness. 
However  much,  and  in  whatever  sense,  it  may  be  con- 

*  Cf.  Mill's  Examination  of   Sir  William    Hamilton's    Philosophy, 
chap.  xvii. 


134:  CONCEPTS  OF  MODERN  PHYSICS. 

tended  that  the  intellect  and  its  object  are  both  real  and 
distinct  entities,  it  can  not  for  a  moment  be  denied  that 
the  object,  of  which  the  intellect  has  cognizance,  is  a 
synthesis  of  objective  and  subjective  elements,  and  is 
thus  primarily,  in  the  very  act  of  its  apprehension  and 
to  the  full  extent  of  its  cognizable  existence,  affected 
by  the  determinations  of  the  cognizing  faculty.  When- 
ever, therefore,  we  speak  of  a  thing,  or  a  property  of  a 
thing,  it  must  be  understood  that  we  mean  a  product  of 
two  factors  neither  of  which  is  capable  of  being  appre- 
hended by  itself.  In  this  sense  all  knowledge  is  said  to 
be  relative. 

2.  Objects  are  known  only  through  their  relations 
to  other  objects.     They  have,  and  can  have,  no  proper- 
ties, and  their  concepts  can  include  no  attributes,  save 
these  relations,  or  rather,  our  mental  representations  of 
them.     Indeed,  an  object  can  not  be  known  or  con- 
ceived otherwise  than  as  a  complex  of  such  relations. 
In  mathematical  phrase :  things  and  their  properties  are 
known  only  as  functions  of  other  things  and  properties. 
In  this  sense,  also,  relativity  is  a  necessary  predicate  of 
all  objects  of  cognition. 

3.  A  particular  operation  of  thought  never  involves 
the  entire  complement  of  the  known  or  knowable  prop- 
erties of  a  given  object,  but  only  such  of  them  as  be- 
long to  a  definite  class  of  relations.     In  mechanics,  for 
instance,  a  body  is  considered  simply  as  a  mass  of  de- 
terminate weight  and  volume  (and  in  some  cases  figure), 
without  reference  to  its   other  physical  or  chemical 
properties.     In  like  manner  each  of  the  several  other 
departments  of  knowledge  effects  a  classification  of  ob- 
jects upon  its  own  peculiar  principles,  thereby  giving 
rise  to  different  series  of  concepts  in  which  each  con- 
cept represents  that  attribute  or  group  of  attributes — 


RELATION  OF  THOUGHTS  TO  THINGS.  '135 

that  aspect  of  the  object — which  it  is  necessary,  -in 
view  of  the  question  in  hand,  to  bring  into  view.  Our 
thoughts  of  things  are  thus,  in  the  language  of  Leib- 
nitz, adopted  by  Sir  "William  Hamilton,  and  after  him 
by  Herbert  Spencer,  symbolical,  not  (or,  at  least,  not 
only)  because  a  complete  mental  representation  of  the 
properties  of  an  object  is  precluded  by  their  number 
and  the  incapacity  of  the  mind  to  hold  them  in  simul- 
taneous grasp,  but  because  many  (and  in  most  cases  the 
greater  part)  of  them  are  irrelevant  to  the  mental  op- 
eration in  progress. 

Again :  the  attributes  comprised  in  the  concept  of 
an  object  being  the  representations  of  its  relations  to 
other  objects,  and  the  number  of  these  objects  being 
unlimited,  it  follows  that  the  number  of  attributes  is 
also  unlimited,  and  that,  consequently,  there  is  no  con- 
cept of  an  object  in  which  its  cognizable  properties  are 
exhaustively  exhibited.  In  this  connection  it  is  worthy 
of  mention  that  the  ordinary  doctrinal  statement  of  the 
relation  of  concepts  to  judgments  is  liable  to  serious 
objection.  A  judgment  is  said  to  be  "  a  comparison  of 
two  notions  (concepts),  with  a  resulting  declaration  of 
their  agreement  or  disagreement "  (Whately),  or  "  a 
recognition  of  the  relation  of  congruence  or  connection 
between  two  concepts  "  (Hamilton).  Here  it  is  assumed 
that  the  concepts  preexist  to  the  act  of  judgment,  and 
that  this  act  simply  determines  the  fact  or  degree  of 
their  congruence  or  confliction.  But  the  truth  is  that 
every  concept  is  the  result  of  a  judgment,  or  of  a  series 
of  judgments,  the  initial  judgment  being  the  recogni- 
tion of  a  relation  between  two  data  of  experience.  In 
most  cases,  indeed,  a  judgment  is  a  collation  of  two 
concepts ;  but  every  synthetic  judgment  (i.  e.,  every 
judgment  in  which  the  predicate  is  more  than  a  mere 


136  CONCEPTS  OF  MODERN  PHYSICS. 

display  of  one  or  more  of  the  attributes  connoted  by 
the  subject)  transforms  both  concepts  which  it  brings 
into  relation,  by  either  amplifying  or  restricting  their 
respective  implications.*  When  a  boy  learns  that  "  a 
whale  is  a  mammal,"  his  notions,  both  of  a  whale  and 
of  a  mammal,  undergo  a  material  change.  From  the 
judgment  of  Thomas  Graham  that  "hydrogen  is  a 
metal,"  both  the  term  "  hydrogen"  and  the  term 
"metal"  emerged  with  new  meanings.  The  announce- 
ment by  Sterry  Hunt,  that  "  just  as  solution  is  chemi- 
cal combination  so  chemical  combination  is  mutual 
solution,"  extended  the  concept  "  solution  "  as  well  as 
the  concept  "  chemical  combination." 

It  is  apparent,  from  these  considerations,  that  the 
concepts  of  a  given  object  are  terms  or  links  in  num- 
berless series  or  chains  of  abstractions  varying  in  kind 
and  diverging  in  direction  with  the  comparisons  insti- 
tuted between  it  and  other  objects ;  that  the  import  and 
scope  of  any  one  of  these  concepts  are  dependent,  not 
only  on  the  number,  but  also  on  the  nature  of  the  rela- 
tions with  reference  to  which  the  classification  of  ob- 

*'  That  this  did  not  escape  the  attention  of  Sir  William  Hamilton,  not- 
withstanding his  definition  of  a  judgment,  appears  from  the  following 
passage  of  his  Lectures  on  Logic  (Am.  ed.,  p.  84 :  "A  concept  is  a 
judgment :  for,  on  the  one  hand  it  is  nothing  but  the  result  of  a  fore- 
gone judgment,  or  series  of  judgments,  fixed  and  recorded  in  a  word,  a 
sign,  and  it  is  only  amplified  by  the  annexation  of  a  new  attribute  through 
a  continuance  of  the  same  process."  Among  German  thinkers  Herbart 
had  a  clear  view  of  the  same  truth.  "  Die  Ausbildung  der  Begriffe,"  he 
says  (Lehrbuch  zur  Psychologic,  §  189,  Werke,  vol.  v,  p.  130),  "ist  der 
langsame,  allmaelige  Erfolg  des  immer  fort  gehenden  Urtheilens."  In 
another  place  (id.  ib.,  §  78,  Werke,  v,  59) :  "  Es  fragt  sich,  ob  die  Begriffe 
im  strengen  logischen  Sinn  nicht  vielmehr  logische  Ideale  seien,  denen 
sich  unser  logisches  Denken  mehr  und  mehr  annaehern  soil.  .  .  .  Es 
wird  sich  ueberdiess  zeigen,  dass  die  Urtheile  es  sind,  wodurch  die  Be- 
griffe dem  Ideal  mehr  und  mehr  angenaehert  werden,  daher  sie  den  letz- 
ten  in  gewissem  Sinne  vorangehen." 


RELATION   OF  THOUGHTS  TO  THINGS.  ^37 

jects  is  effected;  and  that  for  this  reason,  too,  all 
thoughts  of  things  are  fragmentary  and  symbolic  repre- 
sentations of  realities  whose  thorough  comprehension  in  - 
any  single  mental  act,  or  series  of  acts,  is  impossible. 
And  this  is  true,  a  fortiori,  because  the  relations  of 
which  any  object  of  cognition  is  the  entirety,  besides 
being  endless  in  number,  are  also  variable — because,  in 
the  language  of  Herakleitos,  all  things  are  in  a  per- 
petual flux. 

All  metaphysical  or  ontological  speculation  is  based 
upon  a  disregard  of  some  or  all  of  the  truths  here  set 
forth.  /"Metaphysical  thinking  is  an  attempt  to  deduce 
the  true  nature  of  things  from  oar  concepts  of  them. 
Whatever  diversity  may  exist  between  metaphysical 
systems,  they  are  all  founded  upon  the  express  or  im- 
plied supposition  that  there  is  a  fixed  correspondence 
between  concepts  and  their  filiations  on  the  one  hand 
and  things  and  their  modes  of  interdependence  on  the 
other/'  This  fundamental  error  is,  in  great  part,  due  to 
a  delusory  view  of  the  function  of  language  as  an  aid 
to  the  formation  and  fixation  of  concepts.  Roughly 
stated,  concepts  are  the.  meanings  of  words ;  and  the 
circumstance  that  words  primarily  designate  things,  or 
at  least  objects  of  sensation  and  their  sensible  interac- 
tions, has  given  rise  to  certain  fallacious  assumptions 
which,  unlike  the  ordinary  infractions  of  the  laws  of 
logic,  are  in  a  sense  natural  outgrowths  of  the  evolution 
of  thought  (not  without  analogy  to  the  organic  diseases 
incident  to  bodily  life)  and  may  be  termed  structural 
fallacies  of  the  intellect.  These  assumptions  are  : 

1.  That  every  concept  is  the  counterpart  of  a  dis- 
tinct objective  reality,  and  that  hence  there  are  as  many 
things,  or  natural  classes  of  things,  as  there  are  concepts 
or  notions. 


138  CONCEPTS  OF  MODERN  PHYSICS. 

2.  That  the  more  general  or  extensive  concepts  and 
the  realities  corresponding  to  them  preexist  to  the  less 
general,  more  comprehensive  concepts  and  their  corre- 
sponding realities ;  and  that  the  latter  concepts  and  real- 
ities are  derived  from  the  former,  either  by  a  successive 
addition  of  attributes  or  properties,  or  by  a  process  of 
evolution,  the  attributes  or  properties  of  the  former 
being  taken  as  implications  of  those  of  the  latter. 

3.  That  the  order  of  the  genesis  of  concepts  is  iden- 
tical with  the  order  of  the  genesis  of  things. 

4.  That  things  exist  independently  of  and  antece- 
dently to  their  relations ;  that  all  relations  are  between 
absolute  terms ;  and  that,  therefore,  whatever  reality 
belongs  to  the  properties  of  things  is  distinct  from  that 
of  the  things  themselves. 

By  the  aid  of  these  preliminaries  I  hope  to  be  able 
to  assign  to  the  mechanical  theory  its  true  character 
and  position  in  the  history  of  the  evolution  of  thought. 
Before  I  proceed  to  this,  however,  it  may  not  be  with- 
out interest,  in  connection  with  the  preceding  inquiry 
into  the  relation  between  concepts  and  their  correspond- 
ing objects,  to  consider  the  question  which  has  long 
been  the  subject  of  eager  debate,  whether  and  to  what 
extent  conceivability  is  a  test  of  possible  reality.  It  is 
contended  by  J.  S.  Mill  and  his  followers,  that  our  in- 
capacity of  conceiving  a  thing  is  no  proof  of  its  impos- 
sibility ;  while  Whewell  and  Herbert  Spencer  maintain 
(though  not  strictly  in  the  same  sense  and  on  the  same 
grounds)  that  what  is  inconceivable  can  not  be  real  or 
true.*  A  trustworthy  judgment  on  the  merits  of  this 

*  The  precise  form  of  Spencer's  test  of  truth,  which  he  terms  the 
"  Universal  Postulate,"  is  the  "  Inconceivability  of  the  Opposite."  Ex- 
pressed in  the  strict  language  of  logic,  his  thesis  is,  that  every  proposition 
whose  contradictory  is  inconceivable  must  be  true.  But,  inasmuch  as 
every  negation  of  a  proposition  is  the  affirmation  of  its  contradictory, 


KELATION  OF  THOUGHTS  TO  THINGS. 

controversy  can  only  be  formed  after  a  careful  deter- 
mination of  the  conditions  of  conceivability  as  indicated 
by  the  nature  of  the  process  of  conception  which  I  have 
attempted  to  describe. 

It  has  been  shown  that  all  true  conception  consists 
in  the  establishment  of  relations  of  partial  or  total  iden- 
tity between  the  fact  to  be  conceived  and  other  known 
facts  of  experience.  The  first  condition  of  conceiva- 
bility, therefore,  is  that  the  thing  or  phenomenon  in 
question  be  susceptible  of  classification,  i.  e.,  of  total  or  / 
partial  identification  with  objects  or  phenomena  previ- 
ously observed. 

A  second  and  very  obvious  condition  of  conceiva- 
bility is  the  consistency  of  the  elements  of  the  concept 
to  be  formed  with  each  other.  It  is  clear  that  two 
attributes,  one  of  which  is  the  negation  of  the  other, 
can  not  simultaneously  belong  to  the  same  subject  and 
thus  be  parts  of  the  same  concept. 

These  two  are  the  only  conditions  which  are  direct- 
ly deducible  from  the  theory  of  conception,  and  may, 
therefore,  with  some  propriety  be  termed  theoretical 
conditions.  But  there  is  a  third,  practical  condition : 
the  consistency  of  the  new  concept  with  previously- 
formed  concepts  bearing  -upon  the  same  subject-matter. 
As  I  have  said,  this  is  a  practical  condition — not  so 
much  a  condition  of  conceivability  as  of  ready  conceiv- 
ability. For  the  old  concepts  may  be  defective  or  erro- 
neous ;  the  very  concept  with  which  they  conflict  may 
supplement  or  supplant,  rectify  or  destroy  them. 

Now,  it  is  easily  seen  that  fulfillment  of  the  first 
condition  can  not  be  a  test  of  reality.  Facts  or  phe- 
nomena may  present  themselves  to  observation  which  | 

this  is  equivalent  to  the  general  statement  that  whatever  is  inconceivable 
can  not  be  true. 


140  CONCEPTS  OF  MODERN  PHYSICS. 

are  wholly  unlike  any  fact  or  phenomenon  theretofore 
observed,  or  whose  likeness  to  the  prior  data  of  experi- 
ence has  not  yet  been  detected.  The  history  of  science 
is  full  of  startling  discoveries ;  every  period  of  active 
research  brings  to  light  phenomena  which  are  not  only 
unlooked-for,  but  without  apparent  analogy  to  other 
known  facts.  In  view  of  this  Liebig  said  :  "  The  secret 
of  all  those  who  make  discoveries  is  that  they  regard 
nothing  as  impossible."  * 

Thus  far,  then,  I  agree  with  Mr.  Mill.  But  I  can 
not  follow  him  when  he  also  rejects  compliance  with 
the  second  condition  as  a  criterion  of  possibility,  and 
refuses  or  neglects  to  distinguish  between  the  case  of 
inconceivability  by  reason  of  the  apparent  or  real  incon- 
gruity of  a  new  fact  or  phenomenon  with  the  data  of 
past  experience  and  the  very  different  case  of  inconceiv- 
ability on  the  ground  of  inconsistency  between  the  sev- 
eral elements  of  a  proposed  concept.  He  instances  the 
concept  "  a  round  square  "  as  one  which  we  are  unable 
to  form,  and  alleges  that  this  inability  is  due  solely  to 
the  inveteracy  of  our  experience.  "  We  can  not  con- 
ceive a  round  square,"  he  says,f  "  not  merely  because 
no  such  object  has  ever  presented  itself  in  our  expe- 
rience, for  that  would  not  be  enough.  Neither,  for 
anything  we  know,  are  the  two  ideas  in  themselves  in- 
compatible. To  conceive  a  body  all  black  and  yet  all 
white,  would  only  be  to  conceive  two  different  sensa- 
tions as  produced  in  us  simultaneously  by  the  same  ob- 
ject— a  conception  familiar  to  our  experience — and  we 
should  probably  be  as  well  able  to  conceive  a  round 
square  as  a  hard  square,  or  a  heavy  square,  if  it  were 

*  Annalen  der  Pharmacie,  x,  179. 

f  Examination  of   the  Philosophy  of  Sir  William  Hamilton,  i,  88, 
Am.  ed. 


RELATION  OF  THOUGHTS  TO  THINGS. 

not  that  in  our  uniform  experience,  at  the  instant  when 
a  thing  begins  to  be  round,  it  ceases  to  be  square,  so 
that  the  beginning  of  the  one  impression  is  inseparably 
associated  with  the  departure  or  cessation  of  the  other. 
Thus  our  inability  to  form  a  conception  always  arises 
from  our  being  compelled  to  form  another  contradictory 
to  it." 

Our  inability  to  conceive  a  round  square  due  to  the 
fact  "  that  in  our  uniform  experience  at  the  instant 
when  a  thing  begins  to  be  round  it  ceases  to  be  square," 
and  to  the  inseparable  association  between  incipient 
roundness  and  departing  squareness !  "Whether  any  one 
has  ever  had  such  experience  as  is  here  spoken  of,  I  do 
not  know ;  but,  if  he  has,  I  am  confident  that,  even 
after  being  reenforced  by  a  large  inheritance  of  ances- 
tral experience  in  the  light  of  the  modern  theory  of 
evolution,  it  will  prove  insufficient  to  account  for  the 
inseparable  association  which  Mill  brings  into  play. 
The  simple  truth  is,  that  a  round  square  is  an  absurdity, 
a  contradiction  in  terms.  A  square  is  a  figure  bounded 
by  four  equal  straight  lines  intersecting  at  right  angles ; 
a  round  figure  is  a  figure  bounded  by  a  curve ;  and  the 
oldest  definition  of  a  curve  is  that  of  "  a  line  which  is 
neither  a  straight  line  nor  made  up  of  straight  lines." 

Mill's  claim  is,  in  effect,  if  not  in  express  words,  a 
denial  of  the  validity  of  the  laws  of  non-contradiction 
and  excluded  middle,  or  (as  he  himself  would  prefer  to 
say)  an  assertion  that  the  fundamental  laws  of  logic 
are,  like  all  so-called  laws  of  nature,  mere  experiential 
inductions,  uniformity  of  experience  being  their  only 
warrant.  But,  if  these  laws  are  not  absolutely  and  uni- 
versally binding  as  constitutive  principles  of  thought 
and  speech — if  the  same  thing  may,  at  the  same  time, 
be  and  not  be,  and  if  its  affirmation  and  denial  are  not 


14:2  CONCEPTS  OF  MODERN  PHYSICS. 

strict  alternatives— we  are  fairly  landed  in  the  regions 
of  utter  nonsense,  where  all  thinking  is  at  an  end  and 
all  language  without  meaning.  The  laws  in  question 
are  principles  constitutive  of,  because  they  are  tacit 
conventions  preliminary  to,  distinct  thought  and  intelli- 
gible speech  ;  and  they  are  no  more  to  be  suspended  in 
favor  of  Mill's  theory  of  inseparable  association  than  to 
be  abrogated  in  furtherance  of  Hegel's  dialectic  process. 

It  ought  to  be  said  that  there  are  expressions  in  the 
same  chapter  of  Mill's  book,  from  which  I  have  just 
quoted,  which  show  that  the  author  was  very  ill  at  ease 
in  the  presence  of  his  own  theory.  For  instance,  he 
says  :  *  "  These  things  are  literally  inconceivable  to  us, 
our  minds  and  our  experience  being  what  they  are. 
Whether  they  would  be  inconceivable  if  our  minds  were 
the  same,  but  our  experience  different,  is  open  to  dis- 
cussion. A  distinction  may  be  made  which  I  think 
will  be  found  pertinent  to  the  question.  That  the  same 
thing  should  at  once  be  and  not  be — that  identically  the 
same  statement  should  be  both  true  and  false — is  not 
only  inconceivable  to  us,  ~but  we  can  not  conceive  that  it 
could  be  made  conceivable" 

How  strange  that  sentences  like  these  should  come 
from  the  pen  of  John  Stuart  Mill!  First  he  denies 
that  inconceivability  is,  in  any  sense  or  in  any  case,  a 
test  of  truth  or  reality;  but  then  he  says  it  may  be 
otherwise  if  the  inconceivability  itself  is  inconceivable ! 
That  is  to  say :  a  witness  is  utterly  untrustworthy ;  but, 
when  he  makes  a  declaration  respecting  his  own  trust- 
worthiness, he  ought  to  be  believed  ! 

The  whole  theory  of  inseparable  association,  as  here 
advanced  and  applied  by  Mill,  is  simply  groundless,  it 
being  impossible,  under  his  theory,  to  know  what  the 

*  Loc.  cit.,  p.  88. 


RELATION  OF  THOUGHTS  TO  THINGS.  '143 

experience  of  his  numerous  readers  has  been,  except 
again  by  experience  which  he  can  not  have  had,  since 
most  of  these  readers  were  utterly  unknown  to  him. 
And  all  attempts  to  argue  questions  with  any  one  on 
such  a  basis  are  supremely  foolish,  Mill  being  bound,  by 
his  own  doctrine,  to  accept  the  answer,  "  My  experience 
has  been  otherwise,"  as  conclusive.  Mill's  theory  is 
thus  subversive  of  itself,  and  every  earnest  sentence  he 
has  ever  written  is  its  practical  refutation. 

In  reference  to  the  case  of  inconceivability  just  dis- 
cussed,  and  others  analogous  to  it,  it  is  to  be  observed 
that  much  of  the  perplexity  and  confusion  which  is 
characteristic  of  the  disputes  between  Mill  and  his  an- 
tagonists arises  from  the  failure  of  the  disputants  to  dis- 
criminate between  purely  formal  concepts  and  the  men- 
tal representations  of  physical  realities.  There  is  a  very 
wide  distinction  between  the  relation  of  a  concept  to 
the  object  of  thought  in  mathematics,  for  example,  and 
the  corresponding  relation  between  a  concept  of  a  ma- 
terial object  and  that  object  itself.  In  mathematics,  as 
in  all  the  sciences  which  are  conversant  about  single 
relations  or  groups  of  relations  established  (and,  within 
the  limits  of  the  constitutive  laws  of  the  mind,  arbitra- 
rily established)  by  the  mind  itself,  certain  concepts  are 
exhaustive  in  the  sense  that  they  imply,  if  they  do  not 
explicitly  exhibit,  all  the  properties  belonging  to  the 
object  of  thought.  Not  only  the  constituents  of  such 
an  object,  but  also  the  laws  of  their  interdependence, 
being  determined  by  the  intellect,  a  single  concept  may 
be  expanded  into  a  series  of  others.  Thus,  a  parabola 
is  a  line  every  point  in  which  is  equidistant  from  a  fixed 
point  and  a  given  straight  line :  that  is  one  of  its  con- 
cepts. And  in  this  all  the  properties  of  the  parabola — 
that  it  is  a  conic  section  formed  by  cutting  a  cone  par- 


144  CONCEPTS   OF  MODERN  PHYSICS. 

allel  to  one  of  its  sides,  that  the  area  of  any  one  of  its 
segments  is  equal  to  two  thirds  of  its  circumscribed 
rectangle,  etc. — are  implied,  and  from  it  they  may  be 
deduced.  One  of  its  attributes  is  an  implication  of  all 
the  others.  Our  concepts  of  material  objects,  on  the 
contrary,  as  I  have  shown,  are  never  exhaustive,  for 
»  their  complement  of  attributes  is  of  necessity  both  in- 
__complete  and  variable.  To  what  strange  vagaries  this 
confusion  has  given  rise  in  other  departments  of  specu- 
lation we  shall  see  in  a  future  chapter. 

I  come  now  to  the  third  condition  of  conceivability : 
the  consistency  of  the  concept  to  be  formed  with  pre- 
vious concepts  in  pari  materid.  By  far  the  greatest 
number  of  the  cases  of  alleged  inconceivability  are 
traceable  to  a  breach  of  this  condition — to  the  incompat- 
ibility of  new  facts  or  views  with  our  intellectual  pre- 
possessions. Accordingly,  most  of  the  cases  adduced 
by  Mill  in  support  of  his  theory  are  taken  from  this 
class.  But  he  does  not  always  apprehend  their  true 
character,  and  most  of  them  are  very  imperfectly,  if  at 
all,  accounted  for  by  his  theory.  One  of  his  instances 
is  that  of  the  denial,  once  all  but  universal,  of  the  pos- 
sibility of  antipodes,  on  the  ground  of  their  inconceiv- 
ability. According  to  Mill,  this  inconceivability  has 
now  vanished  ;  we  not  only  readily  conceive  them  as 
possible,  but  know  them  to  be  real.  This  is  true  enough ; 
but  it  finds  its  explanation,  not  in  the  law  of  inseparable 
association  to  which  it  is  referred  by  Mill,  but  in  the 
fact  that  our  ancestors  held  an  erroneous  concept  of  the 
action  of  gravity.  They  supposed  that  the  direction  in 
which  gravity  acted  was  an  absolute  direction  in  space ; 
they  did  not  realize  that  it  was  a  direction  toward  the 
earth's  center  of  gravity  ;  downward  to  them  meant 
something  very  different  from  the  sense  we  attach  to 


RELATION   OF   THOUGHTS  TO   THINGS.  '145 

that  word.  With  this  erroneous  concept  they  could  not 
reconcile  the  fact  that  the  force  of  gravity  held  our  an- 
tipodes in  position  as  well  as  ourselves ;  nor  can  we. 
But  we  have  a  juster  concept  of  gravity,  and  the  mode 
and  direction  of  its  action;  the  spurious  notion  with 
which  the  notion  of  antipodes  was  inconsistent  has  been 
removed,  and  the  inconceivability  of  antipodes  is  at  an 
end. 

Similar  observations  apply  to  another  example 
brought  forward  by  Mill :  the  inability  to  conceive 
actio  in  distans,  to  which  extended  reference  has  al- 
ready been  made  in  a  preceding  chapter.  This  inabil- 
ity results  from  the  inconsistency  of  this  cjmcept  with 
the  prevailing  notions  respecting  material  plpesence.  If 
we  reverse  the  proposition  that  a  body  acts  wnere  it  is, 
and  say  that  a  body  is  where  it  acts,  the  inconceivabil- 
ity disappears  at  once.  One  of  the  wisest  utterances  on 
this  subject  is  the  saying  of  Thomas  Carlyle  (quoted  by 
Mill  himself  in  another  place)  :  "  You  say  that  a  body 
can  not  act  where  it  is  not?  With  all  my  heart;  but, 
pray  where  is  it?"  Of  course,  a  reconstitution  of  our 
familiar  concepts  of  material  presence,  in  the  sense  here 
indicated,  would  preclude  the  mechanical  construction 
of  matter  from  elements  absolutely  limited,  hard,  un- 
changeable and  separated  from  each  other  by  absolutely 
void  spaces. 

It  is  hardly  necessary  to  add  that,  generally  speak- 
ing, the  inconceivability  of  a  physical  fact  arising  from 
its  incongruity  with  preconceived  notions  is  no  proof  of 
its  impossibility  or  want  of  reality.  Intellectual  prog- 
ress consists  almost  wholly  in  the  rectification  or  sub- 
version of  old  ideas  not  a  few  of  which  are  held  to  be 
self-evident  during  long  intellectual  periods.  The  in- 
stances already  cited  from  Mill  are  apt  illustrations  of 


146  CONCEPTS  OF  MODERN  PHYSICS. 

• 

this ;  and  they  may  be  cumulated  without  limit.  Until 
the  discovery  of  the  composition  of  water,  of  the  true 
theory  of  combustion,  and  of  the  relative  affinities  of 
potassium  and  hydrogen  for  oxygen,  it  was  impossible 
to  conceive  a  substance  which  would  ignite  on  contact 
with  water,  it  being  one  of  the  recognized  attributes  of 
water — in  other  words,  a  part  of  the  concept  water — 
that  it  antagonized  fire.  This  previous  concept  was 
spurious,  and,  when  it  had  been  destroyed,  the  incon- 
ceivability of  a  substance  like  potassium  disappeared. 
Similarly,  we  are  now  unable  to  conceive  a  warm- 
blooded animal  without  a  respiratory  system,  because 
we  conceive  the  idiothermic  condition  of  an  animal 
organism  to  depend  mainly  on  the  chemical  changes 
taking  place  within  it,  chief  among  which  is  the  oxida- 
tion of  the  blood,  which  requires  some  form  of  contact 
between  the  blood  and  the  air,  and  therefore  some  form 
of  respiration.  If,  however,  future  researches  should 
destroy  this  latter  concept — if  it  should  be  shown  that 
the  heat  of  a  living  body  may  be  produced  in  sufficient 
quantity  by  mechanical  agencies,  such  as  friction — a 
non-respiring  warm-blooded  animal  would  at  once  be- 
come conceivable. 

While  thus  a  physical  phenomenon,  however  little 
we  may  be  able  to  conceive  it  without  violence  to  our 
familiar  ideas,  may  be  real,  it  is  otherwise  in  the  domain 
of  the  formal  sciences,  such  as  logic  and  mathematics. 
There  we  find  concepts  founded  upon  fundamental  post- 
ulates or  axiomatic  truths  with  which  all  new  concepts, 
to  be  valid,  must  be  consistent.  The  fact  is  that,  in  the 
sphere  of  the  ideal  relations  of  space  and  time,  the  third 
condition  of  conceivability  is  at  bottom  identical  with 
the  second,  inasmuch  as  there  all  minor  concepts  are, 
by  implication  at  least,  constituents  of  some  higher,  more 


RELATION  OF  THOUGHTS  TO  THINGS. 

comprehensive  concept  whose  validity  requires  their 
consistency  with  each  other.  All  this  is  equally  true 
of  those  purely  formal  concepts  which  constitute  the 
theoretical  basis  of  some  of  the  physical  sciences,  such 
as  the  general  propositions  of  kinematics  or  phoronom- 
ics ;  within  the  limits  of  their  proper  application  they 
are  authoritative  tests  of  possibility.  And  even  among 
the  physical  truths  based  upon  induction  there  are 
many  whose  universality  is  so  well  established  as  to 
afford  strong,  if  not  conclusive  presumption  against  the 
legitimacy  of  concepts  and  the  reality  of  alleged  phe- 
nomena which  would  invalidate  them. 

The  foregoing  discussion  of  the  question  of  conceiva- 
bility  as  a  test  of  truth  is  by  no  means  exhaustive.  There 
are  topics  connected  with  it  upon  which  it  is  not  my 
province  to  enter.  One  of  these  topics  is  the  specifica- 
tion of  the  conditions  under  which  the  inconsistency 
between  the  elements  of  a  proposed  concept  becomes 
apparent.  In  many  cases  the  inconsistency  is  latent 
and  emerges  only  upon  thorough  exhibition  of  all  the 
implications  of  these  elements  and  their  colligation — 
upon  an  explication  which  is  familiarly  known  as  re- 
ductio  ad  dbsurdum.  The  procedure,  in  such  cases,  is 
in  effect  a  reduction  of  the  propositions  into  which  the 
concept  may  be  resolved  to  their  last  degree  of  homo- 
geneity, so  that  the  conflict  between  them,  if  it  exists, 
becomes  explicit.  The  details  of  this  subject,  however, 
belong  to  treatises  on  logic. 


CHAPTER  X. 

CHARACTER     AND    ORIGIN     OF     THE    MECHANICAL   THEORY. 

ITS     EXEMPLIFICATION     OF     THE     FIRST    AND    SECOND 

RADICAL    ERRORS    OF   METAPHYSICS. 

IT  is  the  distinct  claim  of  modern  physicists  that  the 
mechanical  theory  rests  on  the  sure  foundation  of  sen- 
sible experience,  and  is  thus  contradistinguished  from 
metaphysical  speculation,  which  is  said  (and,  in  the  sense 
indicated  in  the  preceding  chapter,  truly  said)  to  be 
based  on  mere  figments  of  the  intellect.  We  have  now 
arrived  at  a  stage  in  our  discussion  where  the  validity 
of  this  claim  may  be  examined. 

The  mechanical  theory  postulates  mass  and  motion 
as  the  absolutely  real  and  indestructible  elements  of  all 
forms  of  physical  existence.  Ordinarily  these  elements 
are  designated  as  matter  and  force  •  but  this  designa- 
tion is  plainly  inaccurate.  The  action  of  force  upon  a 
body,  in  the  light  of  the  mechanical  theory,  is  simply 
the  transference  of  motion  from  one  body  to  another ; 
force,  in  the  sense  in  which  the  word  is  here  employed, 
is  nothing  else  than  motion  under  the  aspect  of  its  actual 
or  possible  transference.  And  its  necessary  comple- 
ment, or,  rather,  its  essential  correlate — that  which 
would  remain  if  a  body  were  divested  of  everything 
that  is  not  a  form  of  force,  or  mode  of  motion — is  not 
matter^  but  mass. 

Now,  it  is  clear  that  motion  in  itself  is  not,  and  can 


FIRST  AND  SECOND   METAPHYSICAL  ERRORS.     '149 

not  be,  an  object  of  sensible  experience.  We  have  ex- 
periential knowledge  of  moving  bodies,  but  not  of  pure 
motion.  And  it  is  equally  clear  that  mass — or,  to  use 
the  ordinary  term,  inert  matter,  or  matter  per  se — can 
not  be  an  object  of  sensible  experience.  Things  are 
objects  of  sensible  experience  only  by  virtue  of  their 
action  and  reaction.  As  Leibnitz  said,  "  Whatever  does 
not  act  does  not  exist " — quod  non  agit,  non  existlt. 
Mass  is  nothing  whereof  the  senses  have  direct  cogni- 
zance ;  it  is  not  presented  to  them  either  as  volume,  or 
as  solidity,  or  as  impenetrability.  The  only  knowledge 
we  have  of  mass  is  derived  from  the  fact  that  different 
velocities,  or  accelerations,  or  changes  of  motion,  are 
produced  in  different  bodies  (which  may  be  of  the  same 
volume  and  of  the  same  degrees  of  solidity  and  im- 
penetrability) by  the  action  of  the  same  force  or  the 
transference  of  the  same  motion.  Apart  from  the 
atomic  theory,  mass  is  but  another  name  for  inertia  ; 
and  this  is  known,  measured,  and  determined  solely  by 
the  amount  of  force  or  motion  which  must  act  upon,  or 
be  communicated  to,  a  given  body  in  order  to  produce 
in  it  a  determinate  velocity,  or,  more  accurately  and 
generally,  a  determinate  rate  of  acceleration  or  deflec- 
tion. Without  its  relation  to  and  union  with  force 
or  motion,  it  has  no  existence,  just  as  force  or  motion 
has  no  existence  without  its  relation  to  and  union  with 
inertia.  The  reality  of  either  presents  itself  to  experi- 
ence as  well  as  to  thought  only  by  means  of  the  other. 

The  truth  is,  that  neither  mass  nor  motion  is  sub- 
stantially real,  tut  both  are  concepts,  or,  rather,  con- 
stituents of  a  concept — the  concept  matter.  They  are 
ultimate  products  of  generalization — the  intellectual 
vanishing-points  of  the  lines  of  abstraction  which  pro- 
ceed from  the  infimce  species  of  sensible  experience. 


150  CONCEPTS  OF  MODERN  PHYSICS. 

Matter  is  the  summum  genus  of  the  classification  of 
bodies  on  the  basis  of  their  physical  and  chemical  prop- 
erties. It  is  not,  therefore,  a  real  thing,  but  the  ideal 
complement  of  two  attributes  belonging  to  all  bodies 
alike.  The  two  attributes  are  inseparable,  not  only  in 
fact,  but  also  in  thought.  "When,  in  ascending  the  scale 
of  classification,  we  have  progressively  dismissed  from 
our  mental  representations  of  the  several  physical  ob- 
jects all  the  attributes  whereby  they  differ,  we  reach  at 
last  two  attributes  wherein  they  agree,  and  which  can 
not  be  sundered  without  transcending  the  limits  within 
which  the  conception  of  physical  reality  is  possible. 
They  are  both  indispensable  components  of  the  highest 
concept  under  which  any  form  of  physical  existence 
can  be  subsumed. 

From  this  the  true  character  of  the  mechanical  the- 
ory is  at  once  apparent.  That  theory  takes,  not  only 
the  ideal  concept  matter^  but  its  two  inseparable  con- 
stituent attributes,  and  assumes  each  of  them  to  be  a 
distinct  and  real  entity.  And  this  identification  of  con- 
cepts with  real,  sensible  objects,  this  confusion  of  ab- 
stractions with  things,  is  one  of  the  old  fundamental 
errors  of  metaphysical  speculation.  It  is  the  first  of 
the  fallacious  assumptions  of  metaphysics  enumerated 
in  the  last  chapter.*  The  mechanical  theory,  in  com- 
mon with  all  metaphysical  theories,  hypostasizes  par- 
tial, ideal,  and,  it  may  be,  purely  conventional  groups  of 
attributes,  or  single  attributes,  and  treats  them  as  varie- 
ties of  objective  reality.  Its  basis,  therefore,  is  essen- 
tially metaphysical.  The  mechanical  theory  is,  in  fact, 
a  survival  of  mediaeval  realism.  Its  substantial  elements 
are  legitimate  logical  descendants  of  the  universalia 
ante  rem  and  in  re  of  the  scholastics,  differing  from 

*  Supra,  p.  137. 


FIRST  AND  SECOND  METAPHYSICAL  ERRORS. 

them,  at  most,  in  this,  that  they  are  summits  of  abstrac- 
tion reached  by  ascents  along  gradations  of  sensible 
properties  ascertained  by  observation  and  experiment, 
and  not  by  escalades  of  the  misty  heights  of  traditional 
predicables  representing  early,  crude,  and  vague  fancies 
of  the  human  intellect. 

The  metaphysical  character  of  the  mechanical  the- 
ory appears,  however,  not  only  in  its  adoption  of  the 
first  of  the  fallacious  assumptions  of  all  metaphysics, 
according  to  which  each  concept  is  the  counterpart  of  a 
real  thing,  but  also  in  the  second  of  these  assumptions, 
which  is,  as  I  have  said,*  that  the  more  general  or  ex- 
tensive concepts,  and  the  realities  corresponding  to 
them,  preexist  to  the  less  general  and  more  comprehen- 
sive concepts  and  their  corresponding  realities,  and  that 
the  latter  concepts  and  realities  are  derived  from  the 
former  either  by  a  successive  addition  of  attributes  or 
properties,  or  by  a  process  of  evolution,  the  attributes 
or  properties  of  the  former  being  taken  as  implications 
of  those  of  the  latter. 

In  the  leading  metaphysical  systems,  the  order  of 
reality  is  completely  inverted.  The  summa  genera  of 
abstraction — the  highest  concepts — are  deemed  the  most, 
and  the  data  of  sensible  experience  the  least  real  of  all 
forms  of  existence.  The  ground  of  this  fancy  is  that 
the  former,  which  include  the  properties  common  to  all 
things,  are  assumed  to  constitute  their  substance,  i.  e., 
the  permanent,  invariable  substratum  of  the  properties 
by  which  particular  things  are  distinguished,  these 
being  regarded,  by  reason  of  their  variability,  as  mere 
accidents.  According  to  the  older  view  of  the  relation 
of  the  accidents  to  the  substance,  or  of  the  characteristic 
attributes  of  the  lower  to  those  of  the  higher  concepts, 

*  Supra,  p.  13Y  seq. 


152  CONCEPTS  OF  MODERN  PHYSICS. 

the  inferior  concepts  or  realities  are  formed  by  a  suc- 
cessive addition  of  attributes  or  properties  to  the  higher 
concepts  or  realities ;  the  varieties  of  objective  reality 
are  held  to  be  due  to  a  synthesis  of  substance  and  acci- 
dents ;  and  this  view  may,  therefore,  be  called  the  syn- 
thetic view.  In  contrast  to  this  stands  the  later,  ana- 
lytical view,  presented  in  evolutionary  or  pantheistic 
systems  in  which  the  lower  conceptual  or  real  forms  are 
supposed  to  be  contained  or  implied  in  the  higher 
forms  and  to  be  derived  from  them  by  processes  of  evo- 
lution or  development.  All  this  has  its  exact  analogue 
in  the  mechanical  theory.  Forty  years  ago  the  creed 
of  an  ordinary  physicist  was  something  like  this :  Pri- 
mordially  there  existed,  through  an  act  of  creation  or 
from  all  eternity,  myriads  of  hard  and  unchangeable 
material  particles.  There  also  existed  certain  forces 
equally  unchangeable,  such  as  the  forces  of  attraction 
and  cohesion,  heat,  electric,  magnetic,  chemical  forces, 
and  so  on.  To  the  constant  or  variable,  partial  or  con- 
current action  of  these  forces  upon  the  material  parti- 
cles are  due  all  the  phenomena  of  physical  reality.  In 
this  action  the  material  particles  are  the  passive  and  the 
forces  the  active  element ;  but  these  elements,  of  course, 
preexist  to  the  action.  Matter  in  itself  is  passive,  dead ; 
all  motion  or  life  is  caused  by  force ;  and  the  only 
possible  solution  of  the  problems  of  physiology,  no  less 
than  those  of  physics  and  chemistry,  consists  in  the 
enumeration  of  the  forces  acting  upon  the  material 
particles  and  in  the  exact  quantitative  determination  of 
the  effects  produced  by  their  action. 

In  the  main  this  creed  is  evidently  a  reproduction 
of  the  old  synthetic  view  of  metaphysics.  And  it  is 
gradually  giving  way  to  a  new  doctrine  which  is  simi- 
larly a  reproduction  of  the  metaphysical  sequel  which 


FIRST  AND  SECOND  METAPHYSICAL  ERRORS.     '153 

I  have  termed  the  analytical  or  evolutionary  view.  The 
recent  theories  of  the  correlation  and  mutual  converti- 
bility of  forces,  as  part  of  the  principle  of  the  conser- 
vation of  energy,  have  shaken,  if  not  destroyed,  the 
notion  of  a  multiplicity  of  independent  original  forces ; 
and,  moreover,  physiologists  like  Du  Bois-Reymond 
recognize  force  as  the  invariable  concomitant,  if  not  as 
the  essential  attribute  or  primary  quality,  of  matter, 
asserting  that  to  every  constant  primordial  mass  belongs 
a  constant  primordial  quantity  of  force,  and  that  all 
the  transformations  of  matter  are  produced  by  a  differ- 
entiation of  this  primordial  force.  From  this  the  sug- 
gestion is  natural  that  all  the  varieties  of  physical  ex- 
istence were  potentially  contained  in  and  have  been 
gradually  evolved  from  matter  in  general,  or  matter 
per  se. 

In  August,  1874,  Professor  Tyndall,  then  President 
of  the  British  Association,  delivered  an  inaugural  ad- 
dress at  a  meeting  of  the  Association  at  Belfast,  in  which 
he  made  the  following  declaration  : 

"Abandoning  all  disguise,  the  confession  that  I 
feel  bound  to  make  before  you  is  that  I  prolong  the 
vision  backward  across  the  boundary  of  the  experimen- 
tal evidence,  and  discern,  in  that  matter  which  we,  in 
our  ignorance  and  notwithstanding  our  professed  rever- 
ence for  its  Creator,  have  hitherto  covered  with  oppro- 
brium, the  promise  and  potency  of  every  form  and 
quality  of  life." 

This  announcement  gave  rise  to  a  commotion  which 
was  hardly  justified  by  its  tenor.  Eor  the  solemnity  of 
the  avowal  was  somewhat  out  of  proportion  to  its  nov- 
elty. Tyndall's  words  were  little  more  than  a  new 
wording  of  an  old  thought  of  Francis  Bacon,  who  said, 
more  than  two  centuries  ago : 


154  CONCEPTS  OF  MODERN  PHYSICS. 

"  And  matter,  whatever  it  is,  must  be  held  to  be  so 
adorned,  furnished,  and  formed,  that  all  virtue,  essence, 
action,  and  natural  motion  may  be  the 'natural  conse- 
quence and  emanation  thereof."  * 

And  the  same  thing  had  been  repeated,  many  times 
since,  by  the  metaphysical  evolutionists,  in  terms  sub- 
stantially like  those  of  Schelling  :  "  Matter  is  the  gen- 
eral seed-corn  of  the  universe  wherein  everything  is  in- 
volved that  is  brought  forth  in  subsequent  evolution."  f 

Nevertheless,  Tyndall's  statement  is  memorable  and 
significant  as  indicating  the  changes  which  the  mechanical 
theory  is  undergoing  in  the  minds  of  modern  physicists. 

Tyndall  is  one  of  the  most  strenuous  advocates  of  the 
atomo-mechanical  theory  and  a  persistent  stickler  for  its 
dominant  features.  "When  he  speaks  of  matter,  he 
means  a  definite  group  of  distinct  and  real  atoms  or 
molecules.  "Many  chemists  of  the  present  day,"  he 
said  in  another  address  (also  delivered  before  the  British 
Association,  at  Liverpool,  and  republished  by  him 
shortly  before  the  Belfast  meeting  J)  "  refuse  to  speak 
of  atoms  and  molecules  as  real  things.  Their  caution 
leads  them  to  stop  short  of  the  clear,  sharp,  mechani- 
cally-intelligible atomic  theory  enunciated  by  Dalton,  or 
any  form  of  that  theory,  and  to  make  the  doctrine  of 
multiple  proportions  their  intellectual  bourn.  I  respect 
the  caution,  though  I  think  it  is  here  misplaced.  The 
chemists  who  recoil  from  these  notions  of  atoms  and 

"Atque  assercnda  materia  (qualiscunque  ea  sit)  ita  ornata  et  appa- 
rata  et  formata,  ut  omnis  virtus,  essentia,  actus  atque  motus  naturalis  ejus 
consecutio  et  emanatio  esse  possit."  Baco,  De  Princ.  atque  Origg.,  Opp. 
ed.  Bohn,  vol.  ii,  p.  691. 

f  "  Die  Materie  ist  das  allgemeine  Samenkorn  des  Universums,  worin 
Alles  verhuettt  ist,  was  in  spaeteren  EntwicMungen  sich  entfaltet."  Schel- 
ling, Ideen  zu  einer  Philos.  der  Natur,  2d  ed.,  p.  315. 

\  Fragments  of  Science  (Am.  ed.),  p. -358. 


FIRST  AND  SECOND  METAPHYSICAL  ERRORS.        155 

molecules  accept  without  hesitation  the  undulatory  the- 
ory of  light.  Like  you  and  me,  they  one  and  all  believe 
in  an  aether  and  its  light-producing  waves.  Let  us 
consider  what  this  belief  involves.  Bring  your  imagi- 
nation once  more  into  play  and  figure  a  series  of  sound- 
waves passing  through  air.  Follow  them  up  to  their 
origin,  and  what  do  you  there  find?  A  definite,  tan- 
gible, vibrating  body.  It  may  be  the  vocal  chords  of  a 
human  being,  it  may  be  an  organ-pipe,  or  it  may  be  a 
stretched  string.  Follow  in  the  same  manner  a  train 
of  aether-waves  to  their  source ;  remembering  at  the 
same  time  that  your  aether  is  matter,  dense,  elastic,  and 
capable  of  motions  subject  to  and  determined  by  me- 
chanical laws.  What,  then,  do  you  expect  to  find  as 
the  source  of  a  series  of  aether-waves  ?  Ask  your  imagi- 
nation if  it  will  accept  a  vibrating  multiple  proportion 
—Si  numerical  ratio  in  a  state  of  oscillation.*  I  do  not 
think  it  will.  You  can  not  crown  the  edifice  by  this 
abstraction.  The  scientific  imagination,  which  is  here 
authoritative,  demands  as  the  origin  and  cause  of  a  series 
of  aether-waves  a  particle  of  vibrating  matter  quite  as 
definite,  though  it  may  be  excessively  minute,  as  that 
which  gives  origin  to  a  musical  sound.  Such  a  particle 
we  name  an  atom  or  a  molecule.  I  think  the  seeking 
intellect,  when  focused  so  as  to  give  definition  without 
penumbral  haze,  is  sure  to  realize  this  image  at  the  last." 

*  When  Tyndall  wrote  this  he  probably  had  before  him  W.  K.  Clif- 
ford's lecture  delivered  before  the  Royal  Institution  in  1867,  in  which 
occurred  this  passage :  "  In  order  to  explain  the  phenomena  of  light,  it  is 
not  necessary  to  assume  anything  more  than  a  periodical  oscillation  be- 
tween two  states  at  any  given  point  of  space."  (Clifford's  Lectures  and 
Essays,  vol.  i,  p.  85.)  Or  the  suggestion  may  have  been  taken  from  J. 
S.  Mill,  who,  in  a  note  to  chapter  xiv,  book  iii,  of  his  Logic,  referring  to 
certain  observations  of  Dr.  Whewell,  characterizes  the  imponderable 
aether  as  an  "  undulating  agency." 


156  CONCEPTS   OF  MODERN  PHYSICS. 

The  plain  import  of  these  sentences  is,  that  an  sethe- 
real  or  other  atom  or  molecule  is  related  to  its  vibratory 
motion  just  as  any  ordinary  body  is  related  to  its  move- 
ments of  translation — as  a  stellar  or  planetary  body,  for 
instance,  is  related  to  its  movements  of  rotation  or  revo- 
lution ;  and  that  just  as  the  conception  of  the  stellar  or 
planetary  body  of  necessity  precedes  the  conception  of 
its  rotatory  or  revolutionary  motion,  so  also  the  concep- 
tion of  the  atom  or  molecule  of  necessity  precedes  the 
conception  of  the  vibratory  motion  whereof  light,  heat, 
electricity,  chemical  action,  etc.,  are  known,  or  supposed 
to  be,  modes.  In  other  words :  to  make  the  existence 
of  matter,  such  as  we  deal  with  in  action  and  in  thought, 
conceivable,  we  are  constrained,  according  to  Tyndall, 
to  assume  ultimate  material  particles  as  preexisting  to 
those  motions  or  manifestations  of  force  which  are  ap- 
prehended as  light,  heat,  electricity,  chemical  action,  etc. 
And  what  is  true  of  the  concept  is  true  of  the  thing. 
The  thing  must  be,  before  it  can  act  or  be  acted  upon, 
agreeably  to  the  old  maxim  :  Operari  sequitur  esse* 

*  It  requires  but  little  reflection  to  see  that  the  realization  of  definite 
atoms  or  molecules,  susceptible  of,  but  preexisting  to  motion,  in  the 
focus  of  Tyndall's  "  seeking  intellect "  is  sheer  delusion.  Let  us,  for  a 
moment,  contemplate  an  ultimate  particle  of  matter  in  its  state  of  exist- 
ence in  advance  of  all  its  motion.  It  is  without  color,  and  neither  light 
nor  dark  ;  for  color  and  lightness  are,  according  to  the  theory  of  which 
Tyndall  is  a  distinguished  champion,  simply  modes  of  motion.  It  is  simi- 
larly without  temperature — neither  hot  nor  cold,  since  heat,  also,  is  a 
mode  of  motion.  For  the  same  reason  it  is  without  electric,  magnetic 
and  chemical  properties — in  short,  it  is  destitute  of  all  those  qualities  in 
virtue  of  which,  irrespective  of  its  magnitude,  it  could  be  an  appreciable 
object  of  sense,  unless  we  except  the  properties  of  weight  and  extension. 
But  weight  is  a  mere  play  of  attractive  forces ;  and  extension,  too,  is 
known  to  us  only  as  resistance  which,  in  turn,  is  a  manifestation  of  force, 
a  phase  of  motion.  Thus  the  difficulty  in  grasping  these  primordial 
things  lies,  not  in  their  excessive  minuteness,  but  in  their  total  destitu- 
tion of  quality.  The  solid,  tangible  reality  craved  by  Tyndall's  "  scientific 


FIRST  AND   SECOND   METAPHYSICAL  ERRORS.      157 

This  view,  presented  by  Tyndall  in  his  Liverpool 
address,  is  the  old  synthetic  notion  of  metaphysical  re- 
alism. The  atoms  or  molecules  are  the  substances  exist- 
ing in  advance  of  their  different  modes  of  motion  which 
are  superinduced  or  added  to  them  as  their  accidents. 
But  in  the  Belfast  address  this  view  is  (unconsciously, 
no  doubt)  so  modified  as  to  shade  into  the  evolutionary 
or  analytical  aspect.  Matter  is  now  said  to  include  or 
involve  even  the  forms  and  qualities  of  life  at  the  outset 
— to  contain  them,  if  not  actually,  at  least  potentially — so 
that  they  proceed  from  it  by  spontaneous  development. 

That  all  attempts  to  construct  physical  phenomena 
by  a  synthesis  of  hypostasized  conceptual  elements,  un- 
der the  first  or  synthetical  view,  are  futile,  in  physics 
no  less  than  in  metaphysics,  is  now  sufficiently  evident 
upon  considerations  variously  presented.  Whether  these 
elements  be  substance  and  accident,  or  matter  and  force, 
they  are  equally  unreal,  and  no  reality  can  be  produced 
by  their  adjunction.  7  And  the  fancied  evolution  of 
things,  or  lower,  more  comprehensive  concepts  from 
higher,  more  extensive  concepts,  in  conformity  with 
the  second,  analytical  view,  is  also  found  to  be  delusive 
upon  simple  reference  to  the  nature  of  the  process  of 
conception.  Higher  concepts  are  formed  out  of  lower 
concepts  by  the  omission  or  rejection  of  differential  at- 
tributes ;  and  there  is  nothing,  certainly,  in  this  logical 
process  from  which  it  can  be  legitimately  inferred  that 
the  rejected  attributes  are  contained  or  implied  in  those 
that  are  retained  and  in  whose  union  the  higher  con- 
cepts consist. 

imagination  "  is  "  nee  quid,  nee  quantum,  nee  quak,"  and  wholly  vanishes 
from  the  "  seeking  intellect,"  the  moment  this  intellect  attempts  to  seize 
it  apart  from  the  motion  which  is  said  to  presuppose  it  as  its  necessary 
substratum. 


158  CONCEPTS  OF  MODERN  PHYSICS. 

It  is  needless  to  say,  I  trust,  that  this  nowise  affects 
the  validity  of  theories  of  evolution  within  the  domains 
of  real  physical  existence  in  their  application  to  organic 
(and,  within  limits,  to  inorganic)  forms.  Questions  of 
'derivation  and  descent,  and  of  organic  and  functional 
differentiation  and  distribution,  are  questions  of  fact  to 
be  determined  in  accordance  with  the  data  of  observa- 
tion and  experiment.  Modes  of  existence  may  be  ge- 
netically connected,  though  there  is  no  mutual  implica- 
tion of  them,  and  though  no  form  of  physical  reality  is 
legitimately  deducible  from  a  concept.  Aristotle's  dic- 
tum, IK  Be  T&V  vorjrwv  ovbev  ryiverai  /i^e^o?,  has  a  fuller 
meaning  than  that  assigned  to  it  by  his  scholastic  dis- 
ciples :  things  are  not  born  of  concepts.  And,  as  will 
appear  still  more  clearly  in  the  next  chapter,  the  filia- 
tion of  concepts  is  not  'at  all  identical  with  the  filiation 
of  things. 

The  errors  of  evolutionism  in  its  confessedly  meta- 
physical forms  (exhibited  in  numerous  hylozoic  and 
pantheistic  doctrines)  are  more  glaring,  it  is  true,  than 
those  of  materialistic  evolutionism.  It  is  characteristic 
of  many  of  the  most  prominent  metaphysical  systems 
that  the  summa  genera  which  serve  as  the  basis  of  evo- 
lution are  reached  by  leaps  into  vacuity  beyond  the 
boundaries  of  legitimate  generalization.  Thus  Hegel 
evolves  all  things  from  pure  Being,  which,  as  he  him 
self  says,  is  wholly  devoid  of  attributes — a  mere  logical 
phantom  conjured  up  by  a  forced  rejection  of  the  last 
attributes  that  can  be  constitutive  of  the  summum  ge- 
nus of  any  classification  of  phenomena  whatever.*  This 

*  Strictly  speaking,  the  foundation  of  Hegel's  "  dialectic  process "  is 
not  even  a  phantom  of  reality.  "  Being  per  se  "  is  not  so  much  as  the 
mere  locus  of  a  vanished  attribute.  The  copula  between  subject  and 
predicate  is  nothing  more  than  the  formal  expression  of  the  fact  that 


FIRST  AND  SECOND   METAPHYSICAL   ERRORS. 

phantom,  as  Hegel  expressly  declares,  is  not  to  be  dis- 
tinguished from,  and  therefore  identical  with,  pure 
Nothing ;  and  for  this  reason  some  of  Hegel's  intel- 
lectual descendants — Bellingshausen,  Rohmer,  Werder, 
George  and  others — have  boldly  undertaken  to  deduce 
the  phenomenal  world  from  the  alleged  concept  Noth- 
ing or  Zero.  The  same  attempt  is  made  by  other  meta- 
physicians in  whose  systems  the  initial  blank  appears 
under  various  disguises — by  Schopenhauer  and  Hart- 
mann,  for  instance,  whose  germinal  principle  is  an  im- 
personal will,  a  concept  whose  attributes  are  contradic- 
tory of  each  other,  and  which  is,  therefore,  as  void  as  the 
pseudo-concept  Nothing.  The  most  imposing  among 
the  disguises  of  the  substantial  Nothing  as  the  fountain 
and  origin  of  all  phenomenal  existence  are  The  Abso- 
lute and  The  Thing  per  se,  both  of  which  are  denials 
in  terms  of  all  possible  relation,  and  thus  negations  of 
all  possible  attributes,  inasmuch  as  every  attribute  is 
essentially  a  relation.  But,  although  such  concepts  as 
'matter  and  force  are  somewhat  less  hollow  than  the 
pseudo-concepts  of  current  metaphysical  speculations, 
they  are  not  less  unavailable  as  starting-points  for  the 
evolution  of  concrete  physical  realities. 

Like  all  metaphysical  theories,  the  mechanical  the- 
ory, by  its  identification  of  concepts  with  things,  has 
given  rise  to  a  number  of  false  antagonisms  and  ground- 
less discussions.  One  of  the  most  noted  controversies 
of  the  time  is  that  between  the  champions  of  the  me- 

the  relation  of  non-contradiction  or  coexistence  subsists  between  two  at- 
tributes, or  between  an  attribute  and  a  group  of  attributes.  It  is  a  mere 
abstract  line  (or  pair  of  lines)  pointing  from  the  generic  to  the  differen- 
tial constituents  of  a  concept.  "  Pure  Being "  is  simply  the  specter  of 
the  copula  between  an  extinct  subject  and  a  departed  predicate.  It  is  a 
sign  of  predication  which  "lags  superfluous  on  the  stage"  after  both  the 
predicate  and  that  whereof  it  was  predicated  have  disappeared. 


160  CONCEPTS  OF  MODERN  PHYSICS. 

chanical  or  corpuscular  theory  of  matter,  who  assert 
that  it  is  a  real  thing  independent  of  force,  and  the 
defenders  of  the  dynamical  theory,  who  maintain  that 
material  particles  are  mere  centers  or  spheres  of  force. 
The  corpuscular  doctrine  is  held  by  the  majority  of 
physicists  in  common  with  ordinary  men,  while  the 
dynamical  view — originally  the  outgrowth  of  metaphys- 
ical speculation — has  been  broached,  on  grounds  that 
are  alleged  to  be  non-metaphysical,  by  Boscovich,  Am- 
pere, Faraday,  and  many  others.  Faraday's  opinion  is 
concisely  stated  by  Tyndall :  *  "  What  do  we  know  of 
the  atom  apart  from  its  force  ?  You  imagine  a  nucleus 
which  may  be  called  #,  and  surround  it  by  forces  which 
may  be  called  m  /  to  my  mind  the  a  or  nucleus  vanishes 
and  the  substance  consists  of  the  powers  m.  And,  in- 
deed, what  notion  can  we  form  of  the  nucleus  indepen- 
dent of  its  powers  ?  "What  thought  remains  on  which 
to  hang  the  imagination  of  an  a  independent  of  the  ac- 
knowledged forces  ? " 

When  Faraday  reasoned  thus  he  was  probably  una- 
ware that  he  but  reproduced  old  reflections  of  Aristotle  f 
which  have  since  found  frequent  expression  in  the  writ- 
ings of  modern  thinkers,^  of  which  the  following  may 
be  taken  as  an  example  : 

"  It  is  a  mere  delusion  of  the  phantasy  that  some- 
thing, we  know  not  what,  remains  after  we  have  de- 
nuded an  object  of  all  the  predicates  belonging  to  it."  * 

*  Faraday  as  a  Discoverer,  Am.  ed.,  p.   123.     For  Faraday's  own 
statement  of  this  view,  see   his  "Speculation  touching  Electric  Con- 
duction  and   the   Nature   of   Matter,"   Phil.   Mag.,   ser.    iii,  vol.   xxiv, 
p.  136. 

f  DC  Gen.  et  Corrupt.,  ii,  1,  3,  4,  6 ;  Met.,  iii,  5 ;  iv,  2;  vi,  1. 
\  Cf.  i.  a.  Locke,  Essay  on  Human  Understanding,  book  ii,  chapters 
xxiii  and  xxiv. 

*  "  Es  ist  due  blosse  Taeuschung  der  Mnbildungskraft,  doss,  nachdem 


FIRST  AND  SECOND  METAPHYSICAL  ERRORS. 

The  antagonism  thus  presented  is  utterly  baseless. 
Matter  can  no  more  be  realized  or  conceived  as  mere 
passive,  spatial  presence,  than  as  a  concretion  of  forces. 
Force  is  nothing  without  mass,  and  mass  is  nothing 
without  force.  Just  as  the  metaphysician  can  not  con- 
ceive the  "  thing "  or  substance  apart  from  its  proper- 
ties, or,  conversely,  the  properties  apart  from  the  sub- 
stance, so  the  physicist  can  not  grasp  matter  (i.  e.,  mass) 
without  force,  or  force  without  matter.  Mass,  inertia, 
or  matter  per  se^  is  indistinguishable  from  absolute 
nothingness ;  for  mass  reveals  its  presence  or  evinces  its 
reality  only  by  its  action,  its  balanced  or  unbalanced 
force,  its  tension  or  motion.  And,  on  the  other  hand, 
pure  force  is  equally  nothing ;  for  if  we  reduce  the  mass 
upon  which  a  given  force,  however  small,  acts,  to  its 
limit  zero — or,  mathematically  expressed,  until  it  be- 
comes infinitely  small — the  consequence  is  that  the  ve- 
locity of  the  resulting  motion  is  infinitely  great,  and 
that  the  "  thing  "  (if  under  these  circumstances  we  may 
still  speak  of  a  thing)  is  at  any  given  moment  neither 
here  nor  there,  but  everywhere — that  there  is  no  real 
presence.  It  is  impossible,  therefore,  to  construct  mat- 
ter by  a  synthesis  of  forces.  And  it  is  incorrect  to  say, 
with  Bain,*  that  "matter,  force,  and  inertia,  are  the 
three  names  for  substantially  the  same  fact,"  or,  that 
"  force  and  matter  are  not  two  things,  but  one  thing," 
or,  f  that  "  force,  inertia,  momentum,  matter,  are  all  but 
one  fact,"  the  truth  being  that  force  and  inertia  are 
conceptual  integrants  of  matter,  and  neither  is  in  any 
proper  sense  a  fact. 

man  eincm  Object  die  einzigen  Praedikate  die  es  hat,  hinweggenommen  hat, 
noch  Etwas,  man  weiss  nicht  was,  von  ihm  zurueckbleibc"  Schclling, 
Ideen,  etc.,  p.  18. 

*  Logic,  vol.  ii,  p.  225.  f  Ibid.,  p.  389. 


162  CONCEPTS  OF  MODERN  PHYSICS. 

The  radical  fallacy  of  the  corpuscular  as  well  as"  of 
the  dynamical  theory  consists  in  the  delusion  that  the 
conceptual  elements  of  matter  can  be  grasped  as  sep- 
arate and  real  entities.  The  corpuscular  theorists  take 
the  element  of  inertia  and  treat  it  as  real  by  itself, 
while  Boscovich,  Faraday,  and  all  those  who  define 
atoms  or  molecules  as  "  centers  of  force,"  seek  to  real- 
ize the  corresponding  element,  force,  as  an  entity  by 
itself.  In  both  cases  products  of  abstraction  are  mis- 
taken for  kinds  of  reality. 

A  satisfactory  examination  of  the  conceptual  terms 
inertia  and  force,  and  of  their  true  implications,  is  im- 
possible here  without  anticipating  considerations  that 
properly  belong  to  the  following  chapters.  The  essen- 
tial correlation  of  inertia  with  force  is  evinced  by  its 
earliest  definitions.  Newton  expressly  speaks  of  iner- 
tia as  of  a  force.  "  There  is  inherent,"  he  says,*  "  in 
matter  a  force,  a  power  of  resistance,  in  virtue  of  which 
every  body,  as  far  as  in  it  lies,  perseveres  in  a  state  of 
rest  or  of  uniform  rectilinear  motion."  In  the  defini- 
tion since  Newton's  time,  this  mode  of  expression  has 
usually  been  discarded.  Young  f  defines  inertia  as  "  the 
incapability  of  matter  of  altering  the  state  into  which 
it  is  put  by  any  external  cause,  whether  that  state  be 
rest  or  motion ; "  and,  similarly,  Whewell,J  as  "  the 
quantity  of  matter  considered  as  resisting  the  commu- 
nication of  motion."  All  these  definitions  imply,  how- 
ever, that  the  forces  moving  a  body  or  a  particle  as  a 
whole  are  strictly  and  absolutely  extraneous  forces. 
In  the  language  of  Newton,*  force  is  "  impressed  upon 
a  body,  and  exerted  upon  it  to  change  its  state  of  rest 
or  uniform  motion  in  a  straight  line." 

*  Princ.,  Del  iii.  f  Mechanics,  p.  117. 

\  Mechanics,  p.  245.  *  Princ.,  Def.  iv. 


FIRST  AND  SECOND   METAPHYSICAL  ERRORS.      -163 

'*  There  is  little  "difficulty  in  understanding  how  the 
disjunction  of  matter  and  force  and  the  etymological 
import  of  the  word  "inertia"  led  to  the  assumption 
that  matter  is  essentially  passive,  or,  as  it  is  commonly 
expressed,  dead.  When  a  body  is  considered  by  itself 
— conceptually  detached  from  the  relations  which  give 
rise  to  its  attributes — it  is  indeed  inert,  and  all  its  ac- 
tion comes  from  without.  But  this  isolated  existence 
of  a  body  is  a  pure  fiction  of  the  intellect.  Bodies 
exist  solely  in  virtue  of  their  relations;  their  reality 
lies  in  their  mutual  action.  Inert  matter,  in  the  sense 
of  the  mechanical  theory,  is  as  unknown  to  experience 
as  it  is  inconceivable  in  thought.  Every  particle  of 
matter  of  which  we  have  any  knowledge  attracts  every 
other  particle  in  conformity  to  the  laws  of  gravitation  ; 
and  every  material  element  exerts  chemical,  electrical 
and  other  force  upon  other  elements  which,  in  respect 
of  such  force,  are  its  correlates.  A  body  can  not,  in- 
deed, move  itself  ;  but  this  is  true  -for  the  same  reason 
that  it  can  not  exist  in  and  by  itself.  The  very  pres- 
ence of  a  body  in  space  and  time,  as  well  as  its  mo- 
tion, implies  interaction  with  other  bodies,  and  there- 
fore actio  in  distans  ;  consequently  all  attempts  to 
reduce  gravitation  or  chemical  action  to  mere  impact 
#re  aimless  and  absurd.  ** 

Physicists  are  perfectly  aware  that  the  sense  com- 
monly attached  to  the  word  inertia  in  its  application  to 
matter  tf  spurious.  «  The  incapacity  of  all  material 
points,"  says  M.  Poisson,  "  to  put  themselves  in  mo- 
tion, or  to  change  the  motion  which  has  been  commu- 
nicated to  them  without  the  aid  of  a  force,  is  what 
constitutes,  the  inertia  of  matter.  This  word  does  not 
signify  that  -jmtter  -is  incapable  of  action  ;  on  the  con- 
irary,  every  -material  pojjit  at  all  times  finds  the  prin- 


CONCEPTS  OF  MODERN  PHYSICS. 

ciple  of  its  movement  in  the  action  of  other  points,  but 
never  in  itself."  * 

In  spite  of  statements  like  this,  however,  and  not- 
withstanding the  clear  apprehension,  by  leading  physi- 
cists, of  the  true  import  of  the  doctrine  of  inertia,  the 
phantom  of  "  dead  matter  "  incessantly  obtrudes  itself 
anew  as  the  basis  of  cosmological  speculations.  Thus, 
Professor  Philip  Spiller,  the  author  of  a  very  service- 
able manual  of  physics,  and  a  prolific  writer  on  scien- 
tific subjects,  some  years  ago  published  a  cosmologi- 
cal treatise,f  whose  theorems  are  founded  on  the  express 
proposition  that  "  no  material  constituent  of  a  body,  no 
atom,  is  in  itself  originally  endowed  with  force,  but 
that  every  such  atom  is  absolutely  dead,  and  without 
any  inherent  power  to  act  at  a  distance."  $  It  appears 
from  the  further  contents  of  this  treatise  that  he  not 
only  denies  force  to  the  atoms  taken  singly,  but  that 
he  also  denies  the  possibility  of  their  mutual  action. 
He  is  driven,  therefore,  to  the  assertion  of  the  inde- 
pendent substantiality  of  force ;  and,  accordingly,  he 
assumes  force  to  be  an  all  -  pervading  quasi  -  material 
presence — as  he  terms  it,  an  incorporeal  matter  (un- 
koerperlicher  Staff).  In  utter  disregard  of  the  funda- 
mental correlation  of  force  and  mass,  Spiller  identifies 
his  force-substance  with  the  omniferous  sether,  so  that 

*  "  L'impossibilite  ou  sont  tous  les  points  materials  de  se  mettre  en 
mouvement  ou  de  changer  le  mouvement  qui  leur  a  ete  communique", 
sans  le  secours  d'une  force,  est  ce  qu'on  entend  par  Finertie  de  la  mati- 
ere.  Ce  mot  ne  signifie  pas  que  la  matiere  soit  incapable  d'agir ;  car,  au 
contraire,  chaque  point  materiel  trouve  toujours  dans  1'action  d'autres 
points  materiels,  mais  jamais  en  lui  me'me,  le  principe  de  son  mouve- 
ment." Poisson,  Traite  de  Mecanique,  liv.  ii,  chap,  i,  110. 

f  Der  Weltaether  als  kosmische  Kraft.  Berlin,  Denicke's  Verlag, 
1873. 

\  Loc.  cit.,  p.  4. 


FIRST  AND   SECOND   METAPHYSICAL  ERRORS.        165 

this  hypostasized  half -concept,  which,  in  the  view  of 
all  other  physicists,  is  not  only  imponderable,  but  des- 
titute of  cohesive,  chemical,  thermal,  electric,  and  mag- 
netic forces  (which,  indeed,  must  be  destitute  of  them 
if  it  is  to  stand  as  the  mere  substratum  of  these  vari- 
ous modes  of  motion),  and  is,  therefore,  still  more 
"  dead,"  if  possible,  than  ordinary  matter,  now  sudden- 
ly, without  changing  its  name,  and  without  ceasing  to 
be  the  substratum  of  luminar  or  other  undulations, 
comes  to  be  the  very  quintessence  of  all  possible  en- 
ergy. 

Professor  Spiller's  speculations  are  a  strange  revival 
of  the  well-known  dreams  of  Kepler,  who  imagined 
that  the  planets  were  borne  and  carried  along  in  their 
courses  by  an  " irnmateriate  species"  (species  immate- 
riatd)  capable  of  overcoming  the  inertia  of  bodies.* 
Kepler's  "immateriate  species"  is  the  same  wooden 
iron  which  Spiller  exhibits  under  the  name  "  incorpo- 
real matter,"  the  only  difference  being  that  the  absurd- 
ity of  Kepler's  chimera  was  less  glaring  in  the  hazy 
dawn  of  the  mechanical  notions  of  his  time  than  the 
extravagance  of  Spiller's  conceit  in  the  light  of  the 
scientific  atmosphere  of  our  day. 

What  possible  part  Spiller's  dead  matter  could  per- 
form in  any  cosmological  scheme,  it  is  difficult  to  see. 
Unchangeable  particles  destitute  of  gravity  and  all 
other  force,  even  if  the  action  of  force  upon  them  were 
conceivable,  must  be  equally  acted  upon  from  all  sides 
by  the  omnipresent  aether,  and  could  not,  therefore,  in 

*  "  Relinquitur  igitur,  ut  quemadmodum  lux  omnia  terrena  illustrans 
species  est  immateriata  ignis  illius,  qui  est  in  corpore  Solis :  ita  virtus 
hsec,  planetarum  corpora  complexa  et  vehens,  sit  species  immateriata 
ejus  virtutis,  quae  in  ipso  Sole  residet,  inaestimabilis  vigoris,  adeoque 
actus  primus  omnis  motus  mundani,"  etc.  Kepler,  De  Motibus  Stelloe 
Martis,  pars  tertia,  cap.  xxxiii ;  Kepleri  Opp.,  ed.  Frisch,  vol.  iii,  p.  302. 
8 


166  CONCEPTS  OF  MODERN  PHYSICS.. 

any  way  help  to  establish  differences  of  density,  or  other 
differences  not  contained  in,  or  evolvable  from,  the 
83ther  itself.  They  could  not  even  add  to  the  extension 
of  a  body,  much  less  to  its  hardness,  being  wholly  with- 
out the  power  of  resistance ;  but,  waiving  this,  and 
granting  that  extension  without  resistance  is  possible, 
they  would  simply  be  bubbles  of  void  space  encysted  in 
the  universal  sether,  and  to  the  differentiation  of  this 
aether  alone  all  the  phenomena  of  the  material  world 
would  be  due. 

LThe  prevailing  errors  respecting  the  inertia  of  mat- 
ter have  naturally  led  to  corresponding  delusions  as  to 
the  nature  of  force.  Here  we  are  met,  in  limine,  by 
an  ambiguity  in  the  meaning  of  the  term  force  in  phys- 
ics and  mechanics.  When  we  speak  of  a  "  force  of 
nature,"  we  use  the  word  force  in  a  sense  very  different 
from  that  which  it  bears  in  mechanics.  A  "  force  of 
nature,"  is  a  survival  of  ontological  speculation;  in 
common  phraseology  the  term  stands  for  a  distinct  and 
real  entity.  But,  as  a  determinate  mechanical  function, 
force  is  simply  the  rate  of  change  of  momentum — 
mathematically  expressed,  the  differential  of  momen- 
tum at  a  given  instant  of  time.  "Momentum,"  says 
Mr.  Tait,*  "  is  the  time-integral  of  force,  because  force 
is  the  rate  of  change  of  momentum."  In  the  canonical 
text-books  on  physics,  force  is  defined  as  the  cause  of 
motion.  "  Any  cause,"  says  Whewell,f  "  which  moves 
or  tends  to  move  a  body,  or  which  changes  or  tends  to 
change  its  motion,  is  called  force."  So  Clerk  Max- 
well :  £  "  Force  is  whatever  changes  or  tends  to  change 
the  motion  of  a  body  by  a'ltering  either  its  direction  or 
its  magnitude."  Far  greater  insight  into  the  nature  of 

*  On  Some  Recent  Advances  in  Physical  Science,  second  ed.,  p.  347. 
f  Mechanics,  p.  1.  J  Theory  of  Heat,  p.  83. 


FIRST  AND  SECOND  METAPHYSICAL  ERRORS.        167 

force  is  exhibited  in  the  definition  of  Somoff,  though 
the  word  "cause"  is  retained:  "A  material  point  *is 
moved  by  the  presence  of  matter  without  it.  This  ac- 
tion of  extraneous  matter  is  attributed  to  a  cause  which 
is  named  force."  *  Taking  these  definitions  as  correctly 
representing  the  received  theories  of  physical  science, 
jit  is  manifest,  irrespective  of  the  considerations  I  have 
presented  in  this  and  the  preceding  chapters,  that  force 
is  not  an  individual  thing  or  entity  that  presents  itself 
directly  to  observation  or  to  thought,  but  that,  so  far 
as  it  is  treated  as  a  definite  and  unital  term  in  the 
operations  of  thought,  it  is  purely  an  incident  to  the 
conception  of  the  interdependence  of  moving  masses. 
The  cause  of  motion,  or  of  the  change  of  motion,  in  a 
body  is  the.  condition  or  group  of  conditions  upon  which 
the  motion  depends;  and  this  condition  or  group  of 
conditions  is  always  a  corresponding  motion,  or  change 
of  motion,  of  the  bodies  outside  of  the  body  in  question 
which  are  its  dynamical  correlates.^!  Otherwise  ex- 
pressed, force  is  a  mere  inference  from  the  motion 
itself  under  the  universal  conditions  of  reality,  and  its 
measure  and  determination  lie  solely  in  the  effect  for 
which  it  is  postulated  as  a  cause  ;  it  has  no  other  exist- 
ence. The  only  reality  of  force  and  its  action  is  the 
correspondence  between  physical  phenomena  in  con- 
formity with  the  principle  of  the  essential  relativity  of 
all  forms  of  physical  existence. 

^That  force  has  no  independent  reality  is  so  plain 
and  obvious  that  it  has  been  proposed  by  some  thinkers 

*  Somoff,  Theoretische  Mechanik  (trans,  by  Ziwet),  vol.  ii,  p.  155. 

f  "  Der  gcgenwaertig  klar  entwickelte  mechanische  Begriff  der  Kraft," 
says  Zoellner  (Natur  der  Kometen,  p.  323),  "  cnthaelt  nichts  Andera 
als  den  Ausdruck  einer  raeumlichen  und  zeitlichen  Beziehung  zweier 
Koerper." 


168  CONCEPTS  OF  MODERN  PHYSICS. 

to  abolish  the  term  force,  like  the  term  cause,  alto- 
gether, f  However  desirable  a  sparing  use  of  such  terms 
may  be  (as  is  illustrated  in  the  clearness  of  some  mod- 
ern mechanical  treatises  *),  it  is  impracticable  wholly  to 
dispense  with  it,  for  the  reason  that  the  conceptual 
element  force,  when  properly  interpreted  in  terms  of 
experience,  is  a  legitimate  incident  to  the  conception  of 
physical  action,  and,  if  its  name  were  disused,  it  would 
instantly  reappear  under  another  name.  There  are  few 
concepts  which  have  not,  in  science  as  well  as  in  meta- 
physics, given  rise  to  the  same  confusion  that  prevails 
in  regard  to  "  force  "  and  "  cause ; "  and  the  blow  lev- 
eled at  these  would  demolish  all  concepts  whatever. 
Nevertheless,  \it  is  of  the  greatest  moment,  in  all  specu- 
lations concerning  the  interdependence  of  physical  phe- 
nomena, never  to  lose  sight  of  the  fact  that  force  is  a 
purely  conceptual  term,  and  that  it  is  not  a  distinct 
tangible  or  intangible  thingT] 

How  imperfectly  all  this  is  apprehended  in  our 
time  appears  upon  the  most  cursory  examination  of 
elementary  treatises  on  physics  as  well  as  original  sci- 
entific essays.  The  relation  of  force  to  mechanical  mo- 
tion is  incessantly  spoken  of  as  a  "  fact  ascertained  by 
observation  and  verified  by  experiment."  In  an  article 
published  in  July,  1872,  it  is  said :  "  In  regard  to  the 
first  question  (What  produces  motion  ?)  there  is  no  diver- 
sity of  opinion.  All  agree  that  what  produces  change 
or  causes  motion  is  force" f  The  obvious  meaning  of 
this  is,  that  it  might  possibly  admit  of  question  whether 
material  change  or  motion  is  produced  by  force  or  by 

*  Cf.  e.  g.  Kirchhoff,  Vorlesungen  uebcr  mathematischc  Physik.  Hei- 
delberg, 1876. 

f  What  determines  Molecular  Motion,  etc.  By  James  Croll.  Phil. 
Mag.,  fourth  series,  vol.  xl,  p.  37. 


FIRST  AND   SECOND   METAPHYSICAL  ERRORS.     * 

something  else,  and  that  physicists,  on  the  whole,  have 
come  to  the  conclusion  that  it  is  produced  by  force. 
Such  a  question  ought,  indeed,  to  be  gravely  pondered ! 
It  is  like  the  question  which  Mr.  Sachs,  in  his  despair, 
propounded  to  the  world :  "  Who  will  assure  us  that 
the  star  which  the  astronomers  regard  as  Uranus  is 
Uranus  in  fact  ? "  * 

Physicists  generally,  however,  are  in  still  greater 
confusion  as  to  the  nature  of  force  in  another  respect. 
Bodies  are  said  to  be  endowed  with  a  definite  quantity 
of  force ;  it  is  assumed  that  to  every  particular  body  or 
atom  belongs,  or  that  in  such  body  or  atom  is  inherent, 
an  invariable  measure  of  energy.  This  statement,  be- 
sides involving  the  conceit  just  noted  of  the  indepen- 
dent reality  of  force,  implies  the  assumption  that  force 
can  be  an  attribute  or  concomitant  of  a  single  particle 
as  such,  ignoring  the  fact,  which  is  otherwise  well 
known  to  physicists,  that  the  very  conception  of  force 
depends  upon  the  relation  between  two  terms  at  least. 
"  Force,"  says  Clerk  Maxwell,f  "  is  but  one  aspect  of 
that  mutual  action  between  two  bodies  which  is  called 
by  Newton  Action  and  Eeaction,  and  which  is  now 
more  briefly  expressed  by  the  single  word  Stress.* 
And  in  another  place  :  %  "If  we  take  into  account  the 
whole  phenomenon  of  the  action  between  two  portions 
of  matter,  we  call  it  Stress.  .  .  .  But,  if  we  confine  our 
attention  to  one  of  the  portions  of  matter,  we  see,  as  it 
were,  only  one  side  of  the  transaction— namely,  that 
which  affects  the  portion  of  matter  under  our  consider- 
ation— and  we  call  this  aspect  of  the  phenomenon,  with 
reference  to  its  effect,  an  External  Force  acting  on  that 

*  Das  Sonnensystem,  oder  neue  Theorie  vom  Baus  der  Welten,  von 
S.  Sachs,  p.  193  (quoted  by  Fechner). 

f  Matter  and  Motion,  ci.  \  Ib.,  xxxvi,  xxxviii. 


170  CONCEPTS  OF  MODERN  PHYSICS. 

portion  of  matter,  and  with  reference  to*  its  cause  we 
call  it  the  Action  of  the  other  portion  of  matter.  The 
opposite  aspect  of  the  stress  is  called  the  Reaction  on 
the  other  portion  of  matter."  To  the  same  effect  is 
the  statement  of  Rankine:*  "Force  is  an  action  be- 
tween two  bodies  either  causing  or  tending  to  cause 
change  in  their  relative  rest  or  motion."  It  follows 
that  a  "  constant  central  force,"  as  belonging  to  an  indi- 
vidual atom  or  molecule  in  and  by  itself,  is  an  impossi- 
bility. 

*  Applied  Mechanics,  fourth  ed.,  p.  15. 


CHAPTER    XI. 

CHARACTER     AND      ORIGIN     OF    THE    MECHANICAL     THEORY 

(CONTINUED). — rrs    EXEMPLIFICATION  OF  THE    THIRD 

RADICAL    ERROR   OF    METAPHYSICS. 

THERE  are  few  beliefs  which  are  generally  held  to 
be  more  indubitable  than  that  of  the  absolute  solidity 
of  matter.  With  the  exception  of  Descartes  and  his  im- 
mediate followers,  whose  assertion  that  matter  is  noth- 
ing but  extension  is  clearly  indefensible,  philosophers 
and  physicists  alike  have  always  placed  solidity  and  im- 
penetrability in  the  front  rank  of  its  primary  qualities. 
And  this  belief,  in  view  of  the  observed  transforma- 
tions of  material  things,  unavoidably  leads  to  the  doc- 
trine that  matter  consists  of  indivisible,  absolutely  rigid 
particles.  The  opinion  of  Tyndall,  expressed  in  the 
passage  quoted  in  the  last  chapter  from  his  Liverpool 
address,  is  the  opinion  of  the  bulk  of  scientific  men,  as 
well  as  of  persons  without  scientific  training.  To  all 
of  them,  as  to  Tyndall,  it  seems  absurd  to  deny  that 
the  conception  of  matter  necessarily  involves  the  no- 
tion of  definite,  tangible,  and  indestructible  solidity.  It 
is  the  general  tacit  assumption  that,  of  the  three  molec- 
ular states,  or  states  of  aggregation,  in  which  matter 
presents  itself  to  the  senses — the  solid,  the  liquid,  and 
the  gaseous — the  last  two  are  simply  disguises  or  com- 
plications of  the  first ;  that  a  gas,  for  instance,  is  in  fact 
a  group  or  cluster  of  solids,  like  a  cloud  of  dust,  differ- 


172  CONCEPTS  OF  MODERN  PHYSICS. 

ing  from  such  a  cloud  only  by  the  greater  regularity  in 
the  forms  and  distances  of  the  particles  whereof  it  is 
composed,  and  by  the  fact  that  these  particles  are  con- 
trolled in  the  case  of  a  gas  by  their  mutual  attractions 
and  repulsions,  while  in  the  case  of  the  cloud  of  dust 
they  are  under  the  sway  of  extrinsic  forces.  And, 
while  the  transition  of  the  three  molecular  states  into 
each  other  in  regular  and  invariable  order  is  too  obvious 
to  be  ignored,  it  is  supposed  that  the  solid  is  the  pri- 
mary state  of  which  the  liquid  and  gaseous,  or  aeriform, 
states  are  simply  derivatives,  and  that,  if  these  states 
are  considered  as  evolved  the  one  from  the  other,  the 
order  of  evolution  is  from  the  solid  to  the  vapor  or  gas. 
In  this  view  the  solid  form  of  matter  is  not  only  the 
basis  and  origin  of  all  its  further  determinations — of  all 
its  evolutions  and  changes — but  it  is  also  the  true  and 
typical  element  of  its  mental  representation  and  con- 
ception. 

While  this  view  of  the  relation  between  the  molec- 
ular states  of  matter  is  universally  prevalent,  it  is  not 
difficult  to  show  that  it  is  inconsistent  with  the  facts. 
All  evolution  proceeds  from  the  relatively  Indetermi- 
nate to  the  relatively  Determinate,  and  from  the  com- 
paratively Simple  to  the  comparatively  Complex.  And 
(confining  our  attention,  for  the  moment,  to  the  two 
extreme  terms  of  the  evolution,  the  solid  and  the  gas, 
and  ignoring  the  intermediate  liquid)  a  comparison  of 
the  gaseous  with  the  solid  state  of  matter  at  once  shows 
that  the  former  is,  not  the  end,  but  the  beginning  of 
the  evolution.  The  gas  is  not  only  comparatively  in- 
determinate— without  fixity  of  volume,  without  crys- 
talline or  other  structure— but  it  also  exhibits,  in  its 
functional  manifestations,  that  simplicity  and  regularity 
which  are  characteristic  of  all  types  or  primary  forms. 


THIRD  METAPHYSICAL  ERROR.  173 

Looking,  first,  to  the  purely  physical  aspect  of  a  gas — 
I  speak,  of  course,  only  of  gases  which  are  approxi- 
mately perfect,  to  the  exclusion  of  vapors  at  low  tem- 
perature and  of  gases  which  are  readily  coercible :  its 
volume  expands  and  contracts  as  the  pressure  to  which 
it  is  subjected :  its  velocity  of  diffusion  is  inversely 
proportional  to  the  square  root  of  its  density ;  its  rate 
of  expansion  is  uniform  for  equal  increments  of  tem- 
perature ;  its  specific  heat  is  the  same  at  all  tempera- 
tures, and,  in  a  given  weight,  for  all  densities  and  un- 
der all  pressures ;  the  specific  heats  of  equal  volumes 
of  simple  and  incondensible  gases,  as  well  as  of  com- 
pound gases  formed  without  condensation,  are  the  same 
for  all  gases  of  whatever  nature,  and  so  on.  In  all  these 
respects  the  contrast  with  the  liquid  as  well  as  the  solid 
form,  the  relations  of  whose  volumes,  or  structures,  or 
both,  to  temperature  and  to  mechanical  pressure  or 
other  force  are  complicated  in  the  extreme,  is  great  and 
striking.  But  this  contrast  becomes  still  more  signal, 
secondly,  under  the  chemical  aspect.  We  can  not,  in 
any  proper  sense,  assign  the  proportions  of  volume  in 
which  the  combination  of  solids  and  liquids  takes  place 
— indeed,  the  combination  of  solids  as  such  is  impos- 
sible— and  the  numbers  expressive  of  the  proportions  of 
the  combining  weights  upon  their  face  exhibit  an  ap- 
pearance of  irrelation  and  irregularity  which  the  most 
sustained  endeavors  of  scientific  men  (such  as  Dumas, 
Stas,  H.  Carey  Lea,  Cooke,  L.  Meyer,  Mendelejeff, 
Baumhauer)  have  been  unable  to  obliterate.  In  the 
combination  of  gases,  on  the  contrary,  all  is  simplicity 
and  order.  "  The  ratio  of  volumes  in  which  gases  com- 
bine is  always  simple,  and  the  volume  of  the  resulting 
gaseous  product  bears  a  simple  ratio  to  the  volumes  of 
its  constituents  " — such  is  the  law  of  Gay-Lussac.  By 


CONCEPTS  OF  MODERN  PHYSICS. 

weight,  the  ratio  of  combination  between  hydrogen  and 
chlorine  is  1  to  35 -5  ;  by  volumes,  one  volume  of  hy- 
drogen combines  with  one  volume  of  chlorine  (the  vol- 
umes being  taken,  of  course,  at  the  same  pressures  and 
temperatures)  so  as  to  form  two  volumes  of  hydro- 
chloric acid.  Oxygen  and  hydrogen  combine  in  the  pro- 
portion of  16  to  2  by  weight ;  but  one  volume  of  oxy- 
gen combines  with  two  volumes  of  hydrogen,  forming 
two  volumes  of  watery  vapor.  Nitrogen  and  hydrogen, 
whose  atomic  weights,  so  called,  are  14  and  1  respec- 
tively, combine  in  the  simple  ratio  of  one  volume  of  ni- 
trogen to  three  volumes  of  hydrogen,  the  combination 
resulting  in  two  volumes  of  gaseous  ammonia.  And 
carbon,  whose  "atomic  weight"  is  12,  though  it  can 
not  be  actually  obtained  in  gaseous  form,  is  assumed  by 
all  chemists  (for  reasons  not  necessary  to  state  here)  to 
combine  with  hydrogen  in  the  ratio  of  one  volume  to 
four,  so  as  to  yield  two  volumes  of  marsh-gas. 

All  this  warrants  the  conclusion  that/if  there  be  a 
typical  and  primary  state  of  matter,  it  is,  not  the  solid, 
but  the  gas.  And,  this  being  so,  it  follows  that  the 
molecular  evolution  of  matter  conforms  to  the  law  of 
all  evolution  in  proceeding  from  the  indeterminate  to 
the  determinate,  from  the  simple  to  the  complex,  from 
the  gaseous  to  the  solid  form.  Inasmuch,  therefore, 
as  the  explanation  of  any  phenomenon  aims  at  the 
exhibition  of  its  genesis  from  its  simplest  beginnings, 
or  from  its  earliest  f ormsjthe  gaseous  form  of  matter 
is  the  true  basis  for  the  explanation  of  the  solid  form, 
and  not,  conversely,  the  solid  for  the  explanation  of  the 
gas. 

From  the  foregoing  considerations,  I  take  it  to  be 
clear  that  the  true  relation  between  the  molecular  states 
of  matter  is  the  exact  reverse  of  that  universally  as- 


THIRD   METAPHYSICAL  ERROR.  "  175 

* 

sumed.  The  universality  of  this  assumption,  however, 
indicates  that  it  is  due,  not  to  a  mere  chance  error  of 
reasoning,  but  to  some  natural  bias  of  the  mind.  The 
question  arises,  therefore :  What  is  the  origin  of  this 
prevalent  delusion  respecting  the  constitution  of  matter  ? 
I  believe  the  answer  to  this  question  to  be  exceedingly 
simple,  and  important  in  proportion  to  its  simplicity. 
One  of  the  fallacies  to  which  the  human  intellect  is  lia- 
ble by  reason  of  the  laws  of  its  growth,  and  which  I  have 
ventured  to  call  structural  fallacies,  is  that  the  intellect 
tends  to  confound  the  order  of  the  genesis  of  its  ideas 
respecting  material  objects  with  the  order  of  the  genesis 
of  these  objects  themselves.  I  have  heretofore  shown 
that  the  progress  of  our  knowledge  depends  upon  anal- 
ogy— upon  a  reduction  of  the  Strange  and  Unknown  to 
the  terms  of  the  Familiar  and  Known.  In  a  certain 
sense  it  is  true,  what  has  been  often  said,  that  all  cog- 
nition is  recognition.  "  Man  constantly  institutes  com- 
parisons," says  Pott,*  "  between  the  new  which  pre- 
sents itself  to  him  and  the  old  which  he  already  knows." 
That  this  is  so  is  shown  by  the  development  of  lan- 
guage. The  great  agent  in  the  evolution  of  language 
is  metaphor — the  transference  of  a  word  from  its  ordi- 
nary and  received  meaning  to  an  analogous  one.  This 
transference  of  the  name  descriptive  of  a  known  and 
familiar  thing  to  the  designation  of  an  unknown  and 
unfamiliar  thing  typifies  the  proceeding  of  the  intellect 
in  all  cases  where  it  deals  with  new  and  strange  phe- 
nomena. It  assimilates  these  phenomena  to  those  which 
are  known ;  it  identifies  the  Strange,  as  far  as  possible, 
with  the  Familiar  ;  it  apprehends  that  which  is  extraor- 
dinary and  uncommon  in  terms  of  that  which  is  or- 
dinary and  common.  But  that  which  is  most  obvious 

*  Etyraologische  Forschungen,  2d  cd.,  vol.  ii,  p.  139. 


176  CONCEPTS  OF  MODERN  PHYSICS. 

to  the  senses  is  both  the  earliest  and  most  persistent 
presence  in  consciousness,  and  thus  receives  the  stamp 
of  the  greatest  familiarity.  Now,  the  most  obtrusive 
form  of  matter  is  the  solid,  and  for  this  reason  it  is  that 
form  which  is  first  cognized  by  the  infant  intellect  of 
mankind,  and  thus  serves  as  the  basis  for  the  subsequent 
recognition  of  other  forms.  Accordingly  we  find  that, 
on  the  early  stages  of  human  history,  the  solid  alone 
was  apprehended  as  material.  It  was  long  before  even 
atmospheric  air,  obtrusive  as  it  was  in  wind  and  storm, 
came  to  be  known  as  a  form  of  matter.  To  this  day 
words  signifying  wind  or  breath — animus,  spiritus, 
Geist,  ghost,  etc. — are  the  terms  denoting  that  which  is 
the  fundamental  correlate  of  matter,  even  in  the  lan- 
guages of  civilized  nations.  And  it  is  very  questionable 
whether  either  the  ancient  philosophers  or  the  mediae- 
val alchemists  distinctly  apprehended  any  aeriform  sub- 
stance, other  than  atmospheric  air,  as  material.  It  is 
certain  that  up  to  the  time  of  Yan  Helmont,  in  the  latter 
part  of  the  sixteenth  and  the  first  decades  of  the  seven- 
teenth century,  aeriform  matter  was  not  the  subject  of 
sustained  scientific  investigation. 

It  is  obvious  then,  that,  while  the  progress  of  evo- 
lution in  nature  is  from  the  aeriform  to  the  solid  state 
of  matter,  the  progress  of  the  evolution  of  knowledge 
in  the  minds  of  men  was,  conversely,  from  the  solid  to 
the  aeriform;  and,  as  a  consequence,  the  aeriform  or 
gaseous  state  came  to  be  apprehended  as  a  mere  modi- 
fication of  solidity.  For  the  same  reason,  the  first  form 
of  material  action  which  was  apprehended  by  the  dawn- 
ing intellect  of  man  was  the  interaction  between  solids 
— mechanical  interaction— and  from  this,  again,  it  fol-  ' 
lowed  that  the  difference  between  the  solid  and  the  gas 
was  apprehended  as  a  mere  difference  of  distance  be- 


THIRD   METAPHYSICAL  ERROR.  '177 

tween  the  solid  particles,  as  produced  by  mechanical 
motion. 

Again :  familiarity,  in  the  minds  of  ordinary  men, 
is  universally  confounded  with  simplicity.  And,  the 
explanation  of  a  phenomenon  aiming,  as  we  have  seen, 
at  an  exhibition  of  its  genesis  from  its  simplest  be- 
ginnings, the  mind,  in  its  attempts  to  explain  the  gase- 
ous form,  naturally  retraces  the  steps  in  the  evolu- 
tion of  its  ideas  concerning  matter — of  its  concepts  of 
matter — back  to  the  earliest,  most  familiar,  and  there- 
•  fore  apparently  simplest  form  in  which  matter  was  and 
is  apprehended,  and  assumes  the  solid  particle,  the 
atom,  as  the  ultimate  fact,  as  the  primary  element  for 
all  representation  and  conception  of  material  existence. 
[The  assumption  of  the  identity  of  the  order  of  con- 
ception with  the  order  of  reality^\(the  third  of  the  falla- 
cious assumptions  enumerated  in  the  ninth  chapter)[is 
one  of  the  most  fatal  errors  of  ontological  speculation, 
and  has  been  signalized  as  such  by  J.  S.  Mill,  who  fails, 
however,  to  discover  the  true  source  of  this  error  as 
pointed  out  above,  attributing  (as  usual)  the  order  and 
connection  of  our  ideas  to  mere  fortuitous  association. 
"A  large  proportion  of  the  erroneous  thinking,"  he 
says,*  "  which  exists  in  the  world,  proceeds  on  the  tacit 
assumption  that  the  same  order  must  obtain  among  the 
objects  of  nature  which  obtains  among  our  ideas  of 
them."  The  inveteracy  of  this  assumption  and  its  irre- 
pressible dominance  in  ontological  speculation  might  be 
shown  by  numerous  examples.  Spinoza  makes  the  dis- 
tinct declaration  that  "  the  order  and  connection  of  ideas 
are  the  same  as  the  order  and  connection  of  things."  f 

*  Logic,  8th  ed.,  p.  521. 

f  "Ordo  et  connezio  idearum  idem  esl  ac  ordo  ct  connexio  rcrum." 
Eth.  ii,  prop.  7. 


178  CONCEPTS  OF  MODERN  PHYSICS. 

And  even  in  a  late  treatise  on  logic  we  read  that  "  the 
logical  catenation  of  ideas  corresponds  to  the  real  cate- 
nation of  things."  *  Here  again,  then,  the  metaphysical 
character  of  the  atomo-mechanical  theory  becomes  mani- 
fest. 

Although  the  opinion  that  solidity  and  impenetra- 
bility are  not  only  indispensable,  but  also  perfectly 
simple  attributes  of  matter,  is  all  but  universal,  there 
are  some  thinkers  who  do  not  fail  to  see  that  it  is  due 
to  a  prejudice  of  the  intellect.  "  In  the  hypothesis," 
says  M.  Cournot,  f  "to  which  modern  physicists  have 
been  led — that  of  atoms  kept  at  a  distance  from  each 
other,  and  even  at  distances  which,  though  inappreciable 
in  experience,  are  nevertheless  very  great  in  comparison 
with  the  dimensions  of  the  atoms  or  elementary  corpus- 
cles— there  is  nothing  that  compels  the  conception  of 
atoms  as  hard  or  solid  little  bodies  rather  than  as  small, 
soft,  flexible,  or  liquid  masses.  The  preference  which 
we  give  to  hardness  over  softness,  the  tendency  to  rep- 
resent the  atom  or  primordial  molecule  as  a  miniature 
of  a  solid  body,  rather  than  as  a  fluid  mass  of  the  same 
size,  are  therefore  nothing  but  prejudices  of  education 
resulting  from  our  habits  and  the  conditions  of  our 
animal  life.  Consequently  there  is  nothing  more  un- 
founded than  the  old  belief — so  deeply  rooted  in  the 
old  scholastics  and  perpetuated  even  in  modern  doctrines 
—which  makes  impenetrability  added  to  extension  the 
fundamental  property  of  matter  and  of  bodies.  It  is 
too  clear  that  atoms  which  could  never  come  in  contact 
could  much  less  penetrate  each  other,  so  that  the  quality 
said  to  be  fundamental  would,  on  the  contrary,  be  a 

*  "ISenchainement  logique  des  idees  correspond  d  V  enchainement  reel 
des  cfioses."    Delboeuf,  Logique,  p.  91. 

f  De  1'Enchainement,  etc.,  vol.  i,  p.  246  seq. 


THIRD  METAPHYSICAL  ERROR. 

useless,  idle  quality  which  would  never  come  into  play 
and  would  never  be  part  of  the  explanation  of  any 
phenomenon,  and  the  assertion  of  whose  existence  would 
be  gratuitous.  The  same  thing  is  to  be  said  of  extension 
as  an  attribute  or  quality  of  atoms,  inasmuch  as,  upon 
last  analysis  and  in  the  present  state  of  the  sciences,  all 
the  explanations  that  can  be  given  of  physico-chemical 
phenomena  are  perfectly  independent  of  the  hypotheses 
which  can  be  framed  respecting  the  figures  and  dimen- 
sions of  elementary  atoms  or  molecules.  As  to  bodies 
of  finite  dimensions  falling  under  our  senses,  they  are 
all  certainly  penetrable  ;  and  so  far  as  they  are  concerned 
the  continuity  of  forms  of  extension  is  but  an  illusion. 

"  In  the  bodies  that  fall  under  our  senses,  solidity 
and  rigidity,  like  flexibility,  softness,  or  fluidity,  are  so 
many  very  complex  phenomena  which  we  attempt  to 
explain,  as  best  we  can,  by  the  aid  of  hypotheses  re- 
specting the  law  of  the  forces  that  maintain  the  ele- 
mentary molecules  at  (definite)  distances,  and  respecting 
the  extent  of  their  sphere  of  action,  as  compared  with 
the  number  of  molecules  embraced  in  that  sphere  and 
with  the  distances  between  them.  Now,  while  the 
familiar  notion  of  bodies  in  the  solid  state  has  suggested 
the  conception  of  the  rigid  corpuscle  or  elementary 
atom  as  a  philosophical  and  scientific  principle  of  ex- 
planation, there  is  nothing  more  difficult  to  explain  sat- 
isfactorily, by  means  of  the  conception  of  atoms,  than 
the  constitution  of  bodies  in  the  state  of  solidity." 

I  have  already  cited,  in  the  seventh  chapter,  a  pas-" 
sage  of  similar  import  from  the  lectures  of  M.  Cauchy, 
in  which  that  distinguished  mathematician  questions 
the  necessity  of  attributing  to  matter  either  impenetra- 
bility or  extension  (without  which,  or  either  of  which, 
there  can,  of  course,  be  no  solidity)  as  a  primary  quality. 


180  CONCEPTS  OF  MODERN  PHYSICS. 

Solidity,  in  the  sense  in  which  it  is  attributed  to  the 
atom,  is  not  a  fact,  but  the  hypostasis  of  an  abstraction. 
As  M.  Cournot  observes,  an  absolutely  solid  body  is  un- 
known to  experience.  The  consistency  of  the  bodies 
which  present  themselves  to  the  experimental  physicist 
depends  upon  the  preponderance  or  balance  of  forces, 
such  as  the  forces  of  cohesion,  crystallization,  and  heat ; 
and  the  assumption  of  the  absolute  solidity  of  matter 
results  from  that  superficial  and  imperfect  apprehension 
of  the  data  of  sense  (in  conjunction  with  the  disregard 
of  the  essential  relativity  of  all  the  properties  of  things 
to  be  considered  more  at  length  hereafter)  which  is  re- 
flected in  all  the  early  notions  of  mankind. 

The  same  primitive,  perfunctory  and  incomplete 
apprehension  of  the  data  of  sense  has  given  rise  to  the 
further  assumption  that  all  physical  action  is  by  im- 
pact. The  only  interaction  between  bodies  that  is  di- 
rectly appreciable  by  the  senses  of  sight  and  touch  is  a 
change  in  the  state  of  rest  or  motion  by  collision.  A 
thrust  is,  therefore,  the  earliest  and  most  familiar  of 
all  the  observable  actions  of  one  body  upon  another. 
And  when  impact  takes  place  between  two  solids  mov- 
ing with  different  velocities,  or  (what  is  the  same  thing) 
between  a  solid  in  motion  and  another  solid  at  rest,  the 
ordinary  observer  sees  nothing  more  than  a  displace- 
ment of  one  body  by  the  other,  and  a  direct  transfer- 
ence of  motion.  This  displacement  and  transference 
are  taken  to  be  instant  and  the  bodies  are  supposed  to 
be  absolutely  rigid.  But  this  observation  of  the  fact 
is  as  crude  as  its  interpretation  is  inaccurate.  A  more 
careful  study  of  the  phenomena  shows  that  there  is  no 
such  immediate  displacement ;  that  there  is  no  direct 
transference  of  motion ;  that  the  bodies  are  not  abso- 
lutely rigid  ;  that  the  apparently  simple  impact  of  solids 


THIRD   METAPHYSICAL  ERROR.  181 

is  a  very  complex  series  or  group  of  incidents  involving 
not  only  direct  action  and  reaction,  but  also  alternate 
compression  and  expansion,  a  loosening  and  tightening 
of  cohesive  and  crystalline  bonds,  transformations  of 
rectilinear  into  vibratory,  of  molar  into  molecular  mo- 
tion, evolution  and  involution  of  energy — in  short,  mo- 
mentary, if  not  permanent  changes  in  all,  or  nearly  all, 
the  properties  of  bodies  between  which  the  impact  oc- 
curs. In  view  of  this  :  what  does  the  demand  of  the 
atomo-mechanical  theory,  to  admit  no  interaction  be- 
tween bodies  other  than  that  of  impact,  imply  ?  Noth- 
ing less  than  this,  that  the  first  rudimentary  and  unrea- 
soned impressions  of  the  untutored  savage  shall  stand 
for  ever  as  the  basis  of  all  possible  science. 

Suppose  that  Hobbes  had  been  familiar  with  the 
incidents  to  the  origin  and  transformation  of  motion, 
as  they  have  been  brought  to  light  by  observation  and 
experiment  in  recent  times  ;  suppose  he  had  been  able, 
as  clearly  as  Helmholtz  and  Mayer,  or  Thomson  and 
Joule,  to  trace,  not  only  the  rotatory  as  well  as  revolu- 
tionary motions  of  our  planet,  but  also  every  disturb- 
ance upon  it — every  blow  dealt  by  a  living  hand  and 
every  shock  caused  by  the  fall  or  projection  of  an  in- 
animate mass — to  the  undrfferentiated  energy  of  a  pri- 
mordial gaseous  spheroid  from  which  the  sun  and  the 
earth  are  supposed  to  have  been  slowly  precipitated  or 
evolved ;  suppose  that,  whenever  he  observed  the  phe- 
nomenon of  impact  between  two  solids  and  the  apparent 
transference  of  visible  motion  from  one  to  the  other, 
his  thoughts  had  involuntarily  run  back  to  the  embryo 
form  of  this  phenomenon,  the  alternate  contraction  and 
expansion  of  a  formless  mobile  gas:  would  he  have 
written  the  sentence  that  "  there  can  be  no  cause  of 
motion  except  in  a  body  contiguous  and  moved  "  ? 


182  CONCEPTS  OF  MODERN  PHYSICS. 

The  logical  and  mathematical  inadmissibility  of  the 
assumption  of  the  absolute  solidity  of  extended  atoms 
or  molecules  was  pointed  out  in  the  early  part  of  the 
last  century  by  John  Bernoulli,  who  showed  that  it  in- 
volved the  conception  of  an  infinite  power  of  resistance 
to  deformation  or  compression.  And  that  solidity  is 
not  the  simplest,  but  the  most  complicated  phase  of 
material  consistency,  was  urged,  more  than  seventy 
years  ago,  by  Fries,  who  objected  to  all  atomic  theories 
that  "  they  assumed  that  which  is  the  most  difficult, 
viz. :  the  constitution  of  definite  forms,  as  an  original 
datum  and  as  the  starting-point  of  explanation,"  *  where- 
as "  the  great  difficulty  of  the  mathematical  philosophy 
of  nature  is  the  possibility  of  rigid  bodies."  f 

The  absolute  solidity  of  matter  is  one  of  the  forms 
in  which  the  pseudo-concept  of  "  being  per  se "  or 
"  simple  existence  "  is  hypostasized,  in  disregard  of  the 
essential  relativity  of  material  things,  which  I  propose 
to  discuss  in  the  next  chapter. 

*  Fries,  Mathematische  Naturphilosophie  (Heidelberg,  1822),  p.  446. 
f  Id.  ib.,  p.  616.    It  will  be  noticed  that  Fries  here  anticipates  the 
observation  of  Couraot  heretofore  cited. 


CHAPTEE  XII. 

CHARACTER     AND     ORIGIN     OF     THE     MECHANICAL     THEORY 

(CONTINUED). — ITS  EXEMPLIFICATION  OF  THE  FOURTH 

RADICAL    ERROR    OF   METAPHYSICS. 

;/  THE  reality  of  all  things  which  are,  or  can  be,  ob- 
jects of  cognition,  is  founded  upon,  or,  rather,  consists 
in,  their  mutual  relations.  A  thing  in  and  by  itself 
can  be  neither  apprehended  nor  conceived ;  its  exist- 
ence is  no  more  a  presentation  of  sense  than  a  deliver- 
ance of  thought.  Things  are  known  to  us  sglely  through 
their  properties  ;  and  the  properties  of  things  are  noth- 
ing else  than-  their  interactions  and  mutual  relations.' 
"  Every  property  or  quality  of  a  thing,"  says  Helm- 
holtz  *  (speaking  of  the  inveterate  prejudice  according 
to  which  the  qualities  of  things  must  be  analogous  to, 
or  identical  with,  our  perceptions  of  them),  "  is  in  real- 
ity nothing  but  its  capability  of  producing  certain  ef- 
fects on  other  things.  The  effect  occurs  either  between 
like  parts  of  the  same  body  so  as  to  produce  differences 
of  aggregation,  or  it  proceeds  from  one  body  to  an- 
other, as  in  the  case  of  chemical  reactions ;  or  the  ef- 
fects are  upon  our  organs  of  sense  and  manifest  them- 
selves as  sensations  such  as  those  with  which  we  are 
here  concerned  (the  sensations  of  sight).  *  Such  an  effect 
we  call  a  ( property/  its  reagent  being  understood 

*  Die  neueren  Fortschritte  in  der  Thcorie  des  Sehens.     Pop.  wiss. 
Vortraege,  ii,  55  seq. 


1S4:  CONCEPTS  OF  MODERN  PHYSICS. 

without  being  expressly  mentioned.  Thus  we  speak  of 
the  '  solubility'  of  a  substance,  meaning  its  behavior 
toward  water  ;  we  speak  of  its  '  weight,'  meaning  its 
attraction  to  the  earth ;  and  we  may  justly  call  a  sub- 
stance '  blue '  under  the  tacit  assumption  that  we  are 
only  speaking  of  its  action  upon  a  normal  eye.  But, 
if  what  we  call  a  property  always  implies  a  relation 
between  two  things,  then  a  property  or  quality  can 
never  depend  upon  the  nature  of  one  agent  alone,  but 
exists  only  in  relation  to  and  dependence  on  the  nature 
of  some  second  object  acted  upon.  Hence,  there  is 
really  no  sense  in  talking  of  properties  of  light  which 
belong  to  it  absolutely,  independently  of  all  other  ob- 
jects, and  which  are  supposed  to  be  representable  in 
the  sensations  of  the  human  eye.  The  notion  of  such 
properties  is  a  contradiction  in  itself.  They  can  not 
possibly  exisj,  and  therefore  we  can  not  expect  to  find 
any  coincidence  of  our  sensations  of  color  with  quali- 
ties of  light." 

The  truth  which  underlies  these  sentences  is  of  such 
transcendent  importance  that  it  is  hardly  possible  to 
be  too  emphatic  in  its  statement,  or  too  profuse  in  its 
illustration. ' '  The  real  existence  of  things  is  coexten- 
sive with  their  qualitative  and  quantitative  determina- 
tions. And  both  are  in  their  nature  relations,  quality 
resulting  from  mutual  action,  and  quantity  being  sim- 
ply a  ratio  between  terms  neither  of  which  is  absolute. 
Every  objectively  real  thing  is  thus  a  term  in  number- 
less series  of  mutual  implications,  and  forms  of  reality 
beyond  these  implications  are  as  unknown  to  experience 
as  to  thought.  There  is  no  absolute  material  quality, 
no  absolute  material  substance,  no  absolute  physical 
unit,  no  absolutely  simple  physical  entity,  no  absolute 
physical  constant,  no  absolute  standard,  either  of  quan- 


FOURTH  METAPHYSICAL  ERROR.  185 

tity  or  quality,  no  absolute  motion,  no  absolute  rest,  no 
absolute  time,  no  absolute  space.  There  is  no  form  of 
material  existence  which  is  either  its  own  support  or  its 
own  measure,  and  which  abides,  either  quantitatively 
or  qualitatively,  otherwise  than  in  perpetual  change,  in 
an  unceasing  flow  of  mutations.  An  object  is  large 
only  as  compared  with  another  which,  as  a  term  of  this 
comparison,  is  small,  but  which,  in  comparison  with  a 
third  object,  may  be  indefinitely  large ;  and  the  com- 
parison which  determines  the  magnitude  of  objects  is 
between  its  terms  alone,  and  not  between  any  or  all 
of  its  terms  and  an  absolute  standard.  An  object  is 
hard  as  compared  with  another  which  is  soft,  but 
which,  in  turn,  may  be  contrasted  with  a  third  still 
softer ;  and,  again,  there  is  no  standard  object  which  is 
either  absolutely  hard  or  absolutely  soft.  A  body  is 
simple  as  compared  with  the  compound  into  which  it 
enters  as  a  constituent;  but  there  is  and  can  be  no 
physically  real  thing  which  is  absolutely  simple.*^ 

It  may  be  observed,  in  this  connection,  that  not  only 
the  law  of  causality,  the  conservation  of  energy,  and  the 
indestructibility  of  matter,  so  called,  have  their  root  in 
the  relativity  of  all  objective  reality — being,  indeed,  sim- 
ply different  aspects  of  this  relativity — but  that  New- 

*  One  of  the  most  noteworthy  specimens  of  ontological  reasoning  is 
the  argument  which  infers  the  existence  of  absolutely  simple  substances 
from  the  existence  of  compound  substances.  Leibnitz  places  this  argu- 
ment at  the  head  of  his  "  Monadology."  "  Necesse  est,"  he  says,  "  dari 
substantial  simplices  quia  dantur  composites  ;  neque  enim  compositum  est 
nisi  aggregatum  simplicium."  (Leibnitii,  Opera  omnia,  ed.  Dutens,  t. 
ii ,  p.  21.)  But  the  enthymeme  is  obviously  a  vicious  paralogism — a 
fallacy  of  the  class  known  in  logic  as  fallacies  of  suppressed  relative. 
The  existence  of  a  compound  substance  certainly  proves  the  existence  of 
component  parts  which,  relatively  to  this  substance,  are  simple.  But  it 
proves  nothing  whatever  as  to  the  simplicity  of  these  parts  in  them- 
selves. 


186  CONCEPTS  OF  MODERN  PHYSICS. 

ton's  first  and  third  laws  of  motion,  as  well  as  all  laws  of 
least  action  in  mechanics  (including  Gauss's  law  of  move- 
ment under  least  constraint),  are  but  corollaries  from  the 
same  principle.  And  the  fact  that  everything  is,  in  its 
manifest  existence,  but  a  group  of  relations  and  reactions 
at  once  accounts  for  Nature's  inherent  teleology. 

Although  the  truth  that  all  our  knowledge  of  ob- 
jective reality  depends  upon  the  establishment  or  recog- 
nition of  relations  is  sufficiently  evident  and  has  been 
often  proclaimed,  it  has  thus  far  been  almost  wholly 
ignored  by  men  of  science  as  well  as  by  metaphysicians. 
It  is  to  this  day  assumed  by  physicists  and  mathema- 
ticians, no  less  than  by  ontologists,  that  all  reality  is  in 
its  last  elements  absolute.  And  this  assumption  is  all 
the  more  strenuously  insisted  on  by  those  whose  sci- 
entific creed  begins  with  the  proposition  that  all  our 
knowledge  of  physical  things  is  derived  from  expe- 
rience. Thus  the  mathematician,  who  fully  recognizes 
the  validity  of  this  proposition  and  at  the  same  time 
concedes  that  we  have,  and  can  have,  no  actual  knowl- 
edge of  bodies  at  rest  or  in  motion,  except  in  relation 
to  other  bodies,  nevertheless  declares  that  rest  and  mo- 
tion are  real  only  in  so  far  as  they  and  their  elements, 
space  and  time,  are  absolute.  The  physicist  reminds 
us  at  every  step  that  in  the  field  of  his  investigations 
there  are  no  a  priori  truths  and  that  nothing  is  known 
of  the  world  of  matter  save  what  has  been  ascertained 
by  observation  and  experiment ;  he  then  announces  as 
the  uniform  result  of  his  observations  and  experiments, 
that  all  forms  of  material  existence  are  complex  and 
variable  ;  and  yet  he  avers  that  not  merely  the  laws  of 
their  variation  are  constant,  but  that  the  real  constitu- 
ents of  the  material  world  are  absolutely  simple,  inva- 
riable, individual  things. 


FOURTH  METAPHYSICAL  ERROR.  187 

v<The  assumption  that  all  physical  reality  is  in  its  last 
elements  absolute — that  the  material  universe  is  an  ag- 
gregate of  absolutely  constant  physical  units  which  in 
themselves  are  absolutely  at  rest,  but  whose  motion, 
however  induced,  is  measurable  in  terms  of  absolute 
space  and  absolute  time — is  obviously  the  true  logical 
basis  of  the  atomo-mechanical  theory.  And  this  as- 
sumption is  identical  with  that  which  lies  at  the  root  of 
all  metaphysical  systems,  with  the  single  difference  that 
in  some  of  these  systems  the  physical  substratum  of 
motion  (termed  the  "  substance  "of  things)  is  not  spe- 
cialized into  individual  atoms. " 

To  show  how  irrepressibly  the  ontological  prejudice, 
that  nothing  is  physically  real  which  is  not  absolute, 
has  asserted  itself  in  science  during  the  last  three  cen- 
turies, I  propose  briefly  to  review  the  doctrines  of  some 
of  the  most  eminent  mathematicians  and  physicists  re- 
specting space  and  motion  (and,  incidentally,  time),  be- 
ginning with  those  of  Descartes. 

In  the  introductory  parts  of  his  Principia,  Des- 
cartes states  in  the  most  explicit  terms  that  space 
and  motion  are  essentially  relative.  "In  order  that 
the  place  [of  a  body]  may  be  determined,"  he  says,* 
"  we  must  refer  to  other  bodies  which  we  may  regard 
as  immovable,  and  accordingly  as  we  refer  to  differ- 
ent bodies  it  can  be  said  that  the  same  thing  does, 
and  does  not,  change  its  place.  Thus,  when  a  ship  is 
earned  along  at  sea,  he  who  sits  at  the  stern  remains 
always  at  the  same  place  in  reference  to  the  parts 
of  the  ship  among  which  he  retains  the  same  position  ; 
but  he  continually  changes  his  place  in  reference  to  the 
shores.  .  .  .  And  besides,  if  we  allow  that  the  earth 
moves  and  proceeds — precisely  as  far  from  west  to 

*  Princ.  ii,  §  18. 


188  CONCEPTS  OF  MODERN  PHYSICS. 

east  as  the  ship  meanwhile  is  carried  from  east  to 
west — we  shall  say  again  that  he  who  sits  at  the  stern 
does  not  move  his  place,  because  we  determine  it  with 
reference  to  some  immovable  points  in  the  heavens. 
But,  if  finally  we  concede  that  no  truly  immovable 
points  are  to  be  found  in  the  universe,  as  I  shall  here- 
after show  is  probable,  our  conclusion  will  be  that  there 
is  nothing  which  has  a  fixed  place  except  so  far  as  it  is 
determined  in  thought."  * 

Statements  to  the  same  effect  are  found  in  various 
other  parts  of  the  same  book.f  And  of  space  Des- 
cartes does  not  hesitate  to  say  that  is  really  nothing 
in  itself,  and  that  "  void  space "  is  a  contradiction  in 
terms — that,  as  Sir  John  Herschel  puts  it,  J  "  if  it  were 
not  for  the  foot-rule  between  them,  the  two  ends  of  it 
would  be  in  the  same  place."  But,  in  the  further  prog- 
ress of  his  discussions,  having  meanwhile  declared  that 
God  always  conserves  in  the  universe  the  same  quan- 
tity of  motion,  he  all  at  once  takes  it  for  granted  *  that 
motion  and  space  are  absolute  and  therefore  real  en- 
tities. 

This  inconsistency  of  Descartes  is  severely  censured 
by  Leibnitz.  "It  follows,"  says  Leibnitz,]  "that  mo- 
tion is  nothing  but  a  change  of  place,  and  thus,  so  far 

*  The  illustration  of  the  relativity  of  motion  by  the  motion  of  a  ship 
is  of  constant  recurrence  whenever  reference  is  had  to  the  question  dis- 
cussed in  the  text.     Cf .  Leibnitz,  Opp.  ed.  Erdmann,  p.   604 ;  Newton, 
Princ.,   Def.  viii,   Schol.    3;  Euler,   Theoria   Motus   Corporum  Solido- 
rum,  vol.  i,  9,  10;  Berkeley,  Principles  of  Human  Knowledge,  §114; 
Kant,  Metaphysische  Aufansgruende  der  Naturwissenschaft,  Phor.  Grund- 
satz  I ;  Cournot,  De  1'Enchainement,  etc.,  vol.  i,  p.  56 ;  Herbert  Spencer, 
First  Principles,  chapter  iii,  §  17,  etc.,  etc. 

f  E.  g.,  Princ.,  ii,  24,  25,  29,  etc. 
\  Familiar  Lectures,  p.  455. 

*  Princ.,  ii,  §§  37-39. 

fl  Leibn.,  Opp.  math.,  ed.  Gerhardt,  sect,  ii,  vol.  ii,  p.  247. 


FOURTH  METAPHYSICAL  ERROR.  189 

as  phenomena  are  concerned,  consists  in  a  mere  rela- 
tion. This  Cartesius  also  acknowledged ;  but  in  deduc- 
ing his  consequences  he  forgot  his  own  definition  and 
framed  his  laws  of  motion  as  though  motion  were  some- 
thing real  and  absolute"  As  will  be  noticed,  Leibnitz 
here  assumes,  as  a  matter  of  course,  that  what  is  real  is 
also  absolute.  In  view  of  this  it  is  hardly  surprising 
that  he,  too,  falls  into  the  same  inconsistency  with  which 
he  charges  Descartes,  and,  in  his  letters  to  Clarke, 
speaks  of  "  absolutely  immovable  space  "  and  an  "  ab- 
solutely veritable  motion  of  bodies."  * 

Newton,  in  the  great  Scholium  to  the  last  of  the 
"Definitions"  prefixed  to  his  Principia,  sharply  dis- 
tinguishes between  absolute  and  relative  time  and  mo- 
tion. "Absolute  and  mathematical  time,"  he  says,f 
"  in  itself  and  in  its  nature  without  relation  to  anything 
external,  flows  equally  and  is  otherwise  called  duration  ; 
relative,  apparent  and  vulgar  time  is  any  sensible  and 
extrinsic,  accurate  or  unequal  measure  of  duration  by 
motion  which  is  ordinarily  taken  for  true  time.  .  .  . 
Absolute  is  distinguished  from  relative  time  in  astron- 
omy by  the  equation  of  vulgar  time.  For  the  natural 
days,  which  are  vulgarly  taken  in  the  measurement  of 
time  as  equal,  are  unequal.  .  .  .  It  may  ~be  that  there  is 
no  equable  motion  by  which  time  is  accurately  meas- 
ured." J 

"  Absolute  space,  in  its  nature  without  relation  to 
anything  external,  always  remains  similar  and  immova- 
ble ;  of  this  (absolute  space)  relative  space  is  any  mova- 
ble measure  or  dimension  which  is  sensibly  defined  by 
its  place  in  reference  to  bodies,  and  is  vulgarly  taken 

*  Opp.  ed.  Erdmann,  pp.  766,  770. 
f  Princ.  (ed.  Le  Seur  &  Jacq.),  p.  8. 
t  L.  c.,  p.  10. 
9 


£90  CONCEPTS  OF  MODERN  PHYSICS. 

for  immovable  space.*  .  .  .  We  define  all  places  by 
the  distances  of  things  from  some  [given]  body  which 
we  take  as  immovable.  .  .  .  It  may  ~be  that  there  is  no 
body  truly  at  rest  to  which  places  and  motions  are  to 
be  referred?"*  f 

Absolute  motion,  according  to  Newton,  is  "the 
translation  of  a  body  from  one  absolute  place  to  anoth- 
er," and  relative  motion  "the  translation  of  a  body 
from  one  relative  place  to  another.  .  .  .  Absolute  rest 
and  motion  are  distinguished  from  relative  rest  and 
motion  by  their  properties  and  by  their  causes  and 
effects.  It  is  the  property  of  rest  that  bodies  truly  at 
rest  are  at  rest  in  respect  to  each  other.  Hence,  while 
it  is  possible  that  in  the  regions  of  the  fixed  stars,  or 
far  beyond  them,  there  is  some  body  absolutely  at  rest, 
it  is  nevertheless  impossible  to  know  from  the  relative 
places  of  bodies  in  our  regions,  whether  any  such  dis- 
tant body  persists  in  the  given  position,  and  therefore 
true  rest  can  not  be  defined  from  the  mutual  position 
of  these"  [i.  e.,  the  bodies  in  our  regions].  ...  "It  is 
the  property  of  motion  that  the  parts  which  retain  their 
given  positions  to  the  wholes  participate  in  their  motion. 
For  all  the  parts  of  rotating  bodies  tend  to  recede  from 
the  axis  of  motion,  and  the  impetus  of  the  moving 
bodies  arises  from  the  impetus  of  the  parts.  Hence, 
when  the  surrounding  bodies  move,  those  which  move 
within  them  are  relatively  at  rest.  And  for  this  reason, 
true  and  absolute  motion  can  not  be  defined  by  their 
translation  from  the  vicinity  of  bodies  which  are  looked 
upon  as  being  at  rest.\  .  .  .  The  causes  by  which 
true  and  relative  motions  are  distinguished  from  each 
other  are  the  forces  impressed  upon  bodies  for  the 
generation  of  motion.  True  motion  is  generated  or 
*  L.  c.,  p.  9.  f  !*>-,  P- 10.  \  Tb.,  pp.  10, 11. 


FOURTH  METAPHYSICAL  ERROR. 

changed  solely  by  the  forces  impressed  upon  the  body 
moved;  but  relative  motion  may  be  generated  and 
changed  without  the  action  of  forces  upon  it.  For  it 
is  sufficient  that  forces  are  impressed  upon  other  bodies 
to  which  reference  is  had,  so  that  by  'their  giving  way 
a  change  is  effected  in  the  relation  in  which  the  rela- 
tive motion  or  rest  of  the  body  consists.*  .  .  .  The 
effects  by  which  absolute  and  relative  motion  are  mu- 
tually distinguished  are  the  forces  by  which  bodies 
recede  from  the  axis  of  circular  motion.  For  in  purely 
relative  circular  motion  these  forces  are  null,  while  in 
true  and  absolute  motion  they  are  greater  or  less  accord- 
ing to  the  quantity  of  motion."  f 

It  is  apparent  that  in  all  these  definitions  Newton, 
like  Descartes  and  Leibnitz,  assumes  real  motion  to  be 
absolute,  and  that  he  takes  the  terms  relative  motion 
and  apparent  motion  to  be  strictly  synonymous,  not- 
withstanding his  express  admission  (in  the  passages 
which  I  have  italicized)  that  in  fact  there  may  be 
neither  absolute  time  nor  absolute  space.  That  admis- 
sion naturally  leads  to  the  further  admission  that  there 
may  in  fact  be  no  absolute  motion;  but  from  this 
Newton  recoils,  resorting  to  the  expedient  of  trying 
to  find  tenable  ground  for  the  distinction  between 
absolute  and  relative  motion,  despite  the  possible  non- 
existence  of  absolute  time  and  space,  in  what  he  calls 
their  respective  causes  and  effects.  But  these  causes 
and  effects  serve  to  distinguish,  not  relative  from  abso- 
lute change  of  position,  but  simply  change  of  position 
in  one  body  with  reference  to  another  from  simul- 
taneous changes  of  position  in  both  with  reference  to  a 
third. 

Newton's  doctrine  is  pushed  to  its  last  consequences 

*  L.  c.,  p.  11.  f  Ib. 


192  CONCEPTS  OF  MODERN  PHYSICS. 

by  Leonhard  Euler.  In  the  first  chapter  of  his  "  The- 
ory of  the  Motion  of  Solid  or  Kigid  Bodies,"  Euler  be- 
gins with  the  emphatic  declaration  that  rest  and  motion, 
so  far  as  they  are  known  to  sensible  experience,  are 
purely  relative.  'After  referring  to  the  typical  case  of 
the  navigator  in  his  ship,  he  proceeds :  *  "  The  notion 
of  rest  here  spoken  of,  therefore,  is  one  of  relations, 
inasmuch  as  it  is  not  derived  solely  from  the  condition 
of  the  point  O  to  which  it  is  attributed,  but  from  a 
comparison  with  some  other  body  A  ....  And  hence 
it  appears  at  once  that  the  same  body  which  is  at  rest 
with  respect  to  the  body  A  is  in  various  motion  with 
respect  to  other  bodies.  .  .  .  "What  has  been  said  of 
relative  rest  may  be  readily  applied  to  relative  motion ; 
for  when  a  point  O  retains  its  place  with  respect  to  a 
body  A,  it  is  said  to  be  relatively  at  rest,  and,  when  it 
continually  changes  that  place,  it  is  said  to  be  relatively 
in  motion.f  .  .  .  Therefore  motion  and  rest  are  distin- 
guished merely  in  name  and  are  not  opposed  to  each 
other  in  fact,  inasmuch  as  hoth  may  at  the  same  time 
le  attributed  to  the  same  point,  accordingly  as  it  is  re- 
ferred to  different  "bodies.  Nor  does  motion  differ  from 
rest  otherwise  than  as  one  motion  differs  from  an- 
other." \ 

After  thus  insisting  upon  the  essential  relativity  of 
rest  and  motion,  Euler  proceeds,  in  the  second  chapter, 
"On  the  Internal  Principles  of  Motion,"  to  consider 
the  question  whether  or  not  rest  and  motion  are  predi- 
cable  of  a  body  without  reference  to  other  bodies.  To 
this  question  he  unhesitatingly  gives  an  affirmative  an- 
swer, holding  it  to  be  axiomatic  that  "  every  body,  even 
without  respect  to  other  bodies,  is  either  at  rest  or  in 

*  Theoria  motus  Corp.  Sol,  etc.,  cap.  i,  explic.  2. 
f  Ib.t  p.  7.  J:  Jb.,  p.  8. 


FOURTH  METAPHYSICAL  ERROR.  193 

motion,  i.  e.,  is  either  absolutely  at  rest  or  absolutely  in 
motion.*  .  .  .  "  Thus  far,"  he  explains,  "  following  the 
senses,  we  have  not  recognized  any  other  motion  or  rest 
than  that  with  respect  to  other  bodies,  whence  we  have 
called  both  motion  and  rest  relative.  But,  if  we  now 
mentally  take  away  all  bodies  but  one,  and  if  thus  the 
relation  by  which  we  have  hitherto  distinguished  its  rest 
and  motion  is  withdrawn,  it  will  first  be  asked  whether 
or  not  the  conclusion  respecting  the  rest  or  motion  of 
the  remaining  body  still  stands.  For,  if  this  conclusion 
can  be  drawn  only  from  a  comparison  of  the  place  of 
the  body  in  question  with  that  of  other  bodies,  it  fol- 
lows that,  when  these  bodies  are  gone,  the  conclusion 
must  go  with  them.  But,  albeit  we  do  not  know  of  the 
rest  or  motion  of  a  body  except  from  its  relation  to 
other  bodies,  it  is  nevertheless  not  to  be  concluded  that 
these  things  (rest  and  motion)  are  nothing  in  themselves 
but  a  mere  relation  established  by  the  intellect,  and  that 
there  is  nothing  inherent  in  the  bodies  themselves  which 
corresponds  to  our  ideas  of  rest  and  motion.  For, 
although  we  are  unable  to  know  quantity  otherwise 
than  by  comparison,  yet,  when  the  things  with  which 
we  instituted  the  comparison  are  gone,  there  is  still  left 
in  the  body  i\\Q  fundamentum  quantitatis,  as  it  were  ; 
for,  if  it  were  extended  or  contracted,  such  extension 
or  contraction  would  have  to  be  taken  as  a  true  change. 
Thus,  if  but  one  body  existed,  we  should  have  to  say 
that  it  was  either  in  motion  or  at  rest,  inasmuch  as  it 
could  not  be  taken  as  being  both  or  neither.  Whence 
I  conclude  that  rest  and  motion  are  not  merely  ideal 
things,  born  from  comparison  alone,  so  that  there  would 

*  Omne  corpus,  etiam  sine  respectu  ad  alia  corpora,  vel  quiescit  vel 
movelur,  hoc  est,  vel  absolute  quiescit,  vel  absolute  movetur"  lb.,  p.  30 
(cap.  ii,  axioma  7). 


194  CONCEPTS  OF  MODERN  PHYSICS. 

le  nothing  inherent  in  the  body  corresponding  to  them, 
but  that  it  may  be  justly  asked  in  respect  to  a  solitary 
body  whether  it  is  in  motion  or  at  rest.  .  .  .  Inasmuch, 
therefore,  as  we  can  justly  ask  respecting  a  single  body 
itself,  without  reference  to  other  bodies,  or  under  the 
supposition  that  they  are  annihilated,  whether  it  is  at 
rest  or  in  motion,  we  must  necessarily  take  one  or  the 
other  alternative.  But  what  this  rest  or  motion  will 
be,  in  view  of  the  fact  that  there  is  here  no  change  of 
place  with  respect  to  other  bodies,  we  can  not  even 
think  without  admitting  an  absolute  space  in  which  our 
body  occupies  some  given  space  whence  it  can  pass  to 
other  places."  *  Accordingly  Euler  most  strenuously 
insists  on  the  necessity  of  postulating  an  absolute,  im- 
movable space.  "  Whoever  denies  absolute  space,"  he 
says,  "  falls  into  the  gravest  perplexities.  Since  he  is 
constrained  to  reject  absolute  rest  and  motion  as  empty 
sounds  without  sense,  he  is  not  only  constrained  also  to 
reject  the  laws  of  motion,  but  to  affirm  that  there  are 
no  laws  of  motion.  For,  if  the  question  which  has 
brought  us  to  this  point,  What  will  be  the  condition  of 
a  solitary  body  detached  from  its  connection  with  other 
bodies  ?  is  absurd,  then  those  things  also  which  are  in- 
duced in  this  body  by  the  action  of  others  become  un- 
certain and  indeterminable,  and  thus  everything  will 
have  to  be  taken  as  happening  fortuitously  and  without 
any  reason."  f 

That  the  basis  of  all  this  reasoning  is  purely  onto- 
logical  is  plain.  And,  when  the  thinkers  of  the  eigh- 
teenth century  became  alive  to  the  fallacies  of  ontologi- 
cal  speculation,  the  unsoundness  of  Euler's  "  axiom," 
that  rest  and  motion  are  substantial  attributive  entities 
independent  of  all  relation,  could  hardly  escape  their 

*  Theoria  motfts,  etc.,  p.  31.  \  lb.,  p.  32. 


FOURTH  METAPHYSICAL  ERROR.  195 

notice.  Nevertheless,  they  were  unable  to  emancipate 
themselves  wholly  from  Euler's  ontological  preposses- 
sions. They  did  not  at  once  avoid  his  dilemma  by  re- 
pudiating it  as  unfounded — by  denying  that  motion 
and  rest  can  not  be  real  without  being  absolute — but 
they  attempted  to  reconcile  the  absolute  reality  of  rest 
and  motion  with  their  phenomenal  relativity  by  postu- 
lating an  absolutely  quiescent  point  or  center  in  space 
to  which  the  positions  of  all  bodies  could  be  referred. 
Foremost  among  those  who  made  this  attempt  was 
Kant.*  In  the  seventh  chapter  of  his  "  Natural  His- 

(*  It  is  remarkable  how  many  of  the  scientific  discoveries,  specula- 
tions and  fancies  of  the  present  day  are  anticipated  or  at  least  foreshad- 
owed in  the  writings  of  Kant.  Some  of  them  are  enumerated  by  Zoellner 
(Natur  der  Kometen,  p.  455  seq.) — among  them  the  constitution  and 
motion  of  the  system  of  fixed  stars  ;  the  nebular  origin  of  planetary  and 
stellar  systems;  the  origin,  constitution  and  rotation  of  Saturn's  rings 
and  the  conditions  of  their  stability ;  the  non-coincidence  of  the  moon's 
center  of  gravity  with  her  center  of  figure ;  the  physical  constitution  of 
the  comets ;  the  retarding  effect  of  the  tides  upon  the  rotation  of  the 
earth ;  the  theory  of  the  winds,  and  Dove's  law.  Fritz  Schultze  has 
shown  (Kant  and  Darwin,  Jena,  1875)  that  Kant  was  one  of  the  pre- 
cursors of  Darwin.  In  this  connection  it  is  curious  to  note  a  coincidence 
(no  doubt  wholly  accidental)  in  the  example  resorted  to  both  by  Kant 
and  A.  R.  Wallace  for  the  purpose  of  illustrating  "  adaptation  by  general 
law."  The  case  put  by  both  is  that  of  the  channel  of  a  river  which,  in 
the  view  of  the  teleologists,  as  Wallace  says  (Contributions  to  the 
Theory  of  Natural  Selection,  p.  276  seq.),  "must  have  been  designed,  it 
answers  its  purpose  so  effectually,"  or,  as  Kant  expresses  it,  must  have 
been  scooped  out  by  God  himself.  ("  Wenn  man  die  physisch-theolo- 
gischen  Verfasser  boert,  so  wird  man  dahin  gebracht,  sich  vorzustellen, 
ihre  Lanfrinnen  waeren  alle  von  Gott  ausgehoehlt."  Beweisgrund 'zu 
einer  Demonstration  des  Dasein's  Gottes,  Kant's  Werke,  i,  p.  232.) 
Even  of  the  vagaries  of  modem  transcendental  geometry  there  are  sug- 
gestions in  Kant's  essays,  Von  dcr  wahren  Schaetzung  der  lebendigen 
Kraefte,  Werke  v,  p.  5,  and  Von  dem  ersten  Grunde  des  Unterschiedes 
der  Gegenden  im  Raume,  ib.,  p.  293— a  fact  which  is  not  likely  to  con- 
duce to  the  edification  of  those  who,  like  J.  K.  Becker,  Tobias,  Weissen- 
born,  Krause,  etc.,  have  raised  the  Kantian  standard  in  defense  of  Eu- 


196  CONCEPTS  OF  MODERN  PHYSICS. 

tory  of  the  Heavens  " — the  same  work  in  which,  nearly 
fifty  years  before  Laplace,  he  gave  the  first  outlines  of 
the  Nebular  Hypothesis — he  sought  to  show  that  in  the 
universe  there  is  somewhere  a  great  central  body  whose 
center  of  gravity  is  the  cardinal  point  of  reference  for 
the  motions  of  all  bodies  whatever.  "If  in  the  im- 
measurable space,"  he  says,*  "  wherein  all  the  suns  of 
the  milky  way  have  been  formed,  a  point  is  assumed 
round  which,  from  whatever  cause,  the  first  formative 
action  of  nature  had  its  play,  then  at  that  point  a  body 
of  the  largest  mass  and  of  the  greatest  attractions,  must 
have  been  formed.  This  body  must  have  become  able 
to  compel  all  systems  which  were  in  process  of  forma- 
tion in  the  enormous  surrounding  sphere  to  gravitate 
toward  it  as  their  center,  so  as  to  constitute  an  entire 
system,  similar  to  the  solar  and  planetary  system  which 
was  evolved  on  a  small  scale  out  of  elementary  mat- 
ter." 

A  suggestion  similar  to  that  of  Kant  has  recently 
been  made  by  Professor  C.  Neumann,  who  enforces  the 
necessity  of  assuming  the  existence,  at  a  definite  and 
permanent  point  in  space,  of  an  absolutely  rigid  body, 
to  whose  center  of  figure  or  attraction  all  motions  are 
to  be  referred,  by  physical  considerations.  The  drift 
of  his  reasoning  appears  in  the  following  extracts  from 
his  inaugural  lecture  On  the  Principles  of  the  Galileo- 

klidean  space.  It  is  probably  not  without  significance  that  in  the  sec- 
ond edition  of  his  Critique  of  Pure  Reason  Kant  omits  the  third  para- 
graph of  the  first  section  of  the  Transcendental  Aesthetics,  in  which 
he  had  enforced  the  necessity  of  assuming  the  a  priori  character  of  the 
idea  of  space  by  the  argument  that  without  this  assumption  the  proposi- 
tions of  geometry  would  cease  to  be  true  apodictically, and  that  "all  that 
could  be  said  of  the  dimensions  of  space  would  be  that  thus  far  no  space 
had  been  found  which  had  more  than  three  dimensions." 
*  Naturgeschichte  des  Himmels,  Werke,  vol.  vi,  p.  152. 


FOURTH  METAPHYSICAL  ERROR.  197 

Newtonian  Theory  :  *  "  The  principles  of  the  Galileo- 
Newtonian  theories  consist  in  two  laws — the  law  of 
inertia  proclaimed  by  Galileo,  and  the  law  of  attraction 
added  by  Newton.  ...  A  material  point,  when  once 
set  in  motion,  free  from  the  action  of  an  extraneous 
force,  and  wholly  left  to  itself,  continues  to  move  in  a 
straight  line  so  as  to  describe  equal  spaces  in  equal 
times.  Such  is  Galileo's  law  of  inertia.  It  is  impossi- 
ble that  this  proposition  should  stand  in  its  present  form 
as  the  corner-stone  of  a  scientific  edifice,  as  the  starting- 
point  of  mathematical  deductions.  For  it  is  perfectly 
unintelligible,  inasmuch  as  we  do  not  know  what  is 
meant  by  '  motion  -in  a  straight  line,'  or,  rather,  inas- 
much as  we  do  know  that  the  words  l  motion  in  a  straight 
line '  are  susceptible  of  various  interpretations.  A  mo- 
tion, for  instance,  which  is  rectilinear  as  seen  from  the 
earth,  would  be  curvilinear  as  seen  from  the  sun,  and 
would  be  represented  by  a  different  curve  as  often  as 
we  change  our  point  of  observation  to  Jupiter,  to  Sat- 
urn, or  another  celestial  body.  In  short,  every  motion 
which  is  rectilinear  with  reference  to  one  celestial  body 
will  appear  curvilinear  with  reference  to  another  celes- 
tial body.  .  .  . 

"  The  words  of  Galileo,  according  to  which  a  mate- 
rial point  left  to  itself  proceeds  in  a  straight  line,  appear 
to  us,  therefore,  as  words  without  meaning — as  express- 
ing a  proposition  which,  to  become  intelligible,  is  in 
need  of  a  definite  background.  There  must  ~be  given  in 
the  universe  some  special  hody  as  the  basis  of  our  com- 
parison, as  the  object  in  reference  to  which  all  motions 
are  to  he  estimated ;  and  only  when  such  a  body  is 
given  shall  we  be  able  to  attach  to  those  words  a  defi- 

*  Ueber  die  Principien  der  Galileo-Newton'schen  Theorie,     Leipzig, 
B.  G.  Teubner,  1870. 


198  CONCEPTS  OF  MODERN  PHYSICS. 

nite  meaning.  Now,  what  body  is  it  which,  is  to  occupy 
this  eminent  position  ?  Or,  are  there  several  such  bod- 
ies ?  Are  the  motions  near  the  earth  to  be  referred  to 
the  terrestrial  globe,  perhaps,  and  those  near  the  sun  to 
the  solar  sphere  ?  .  .  . 

"  Unfortunately,  neither  Galileo  nor  Newton  gives 
us  a  definite  answer  to  this  question.  But,  if  we  care- 
fully examine  the  theoretical  structure  which  they  erect- 
ed, and  which  has  since  been  continually  enlarged,  its 
foundations  can  no  longer  remain  hidden.  We  readily 
see  that  all  actual  or  imaginable  motions  in  the  universe 
must  be  referred  to  one  and  the  same  'body.  Where 
this  body  is,  and  what  are  the  reasons  for  assigning  to 
it  this  eminent,  and,  as  it  were,  sovereign  position,  these 
are  questions  to  which  there  is  no  answer. 

"  It  will  l)e  necessary r,  therefore,  to  establish  the  prop- 
osition^ as  the  first  principle  of  the  Galileo-Newtonian 
theory,  that  in  some  unknown  place  of  the  universe  there 
is  an  unknown  body — a  body  absolutely  rigid  and  un- 
changeable for  all  time  in  its  figure  and  dimensions. 
I  may  ~be  permitted  to  call  this  body  '  THE  BODY  ALPHA.' 
It  would  then  be  necessary  to  add  that  the  motion  of  a 
body  would  import,  not  its  change  of  place  in  reference 
to  the  earth  or  sun,  but  its  change  of  position  in  refer- 
ence to  the  body  Alpha. 

"  From  this  point  of  view  the  law  of  Galileo  is  seen 
to  have  a  definite  meaning.  This  meaning  presents  it- 
self as  a  second  principle,  which  is,  that  a  material  point 
left  to  itself  progresses  in  a  straight  line — proceeds, 
therefore,  in  a  course  which  is  rectilinear  in  reference 
to  the  body  Alpha." 

After  thus  showing,  or  attempting  to  show,  that  the 
reality  of  motion  necessitates  its  reference  to  a  rigid 
body  unchangeable  in  its  position  in  space,  Neumann 


FOURTH  METAPHYSICAL  ERROR.  199 

seeks  to  verify  this  assumption  by  asking  himself  the 
question,  what  consequences  would  ensue,  on  the  hy- 
pothesis of  the  mere  relativity  of  motion,  if  all  bodies 
but  one  were  annihilated.  "  Let  us  suppose,"  he  says, 
"  that  among  the  stars  there  is  one  which  consists  of 
fluid  matter,  and  which,  like  our  earth,  is  in  rotatory 
motion  round  an  axis  passing  through  its  center.  In 
consequence  of  this  motion,  by  virtue  of  the  centrifugal 
forces  developed  by  it,  this  star  will  have  the  form  of 
an  ellipsoid.  What  form,  now,  I  ask,  will  this  star  as- 
sume if  suddenly  all  other  celestial  bodies  are  annihi- 
lated ? 

"  These  centrifugal  forces  depend  solely  upon  the 
state  of  the  star  itself ;  they  are  wholly  independent  of 
the  other  celestial  bodies.  These  forces,  therefore,  as 
well  as  the  ellipsoidal  form,  will  persist,  irrespective  of 
the  continued  existence  or  disappearance  of  the  other 
bodies.  But,  if  motion  is  defined  as  something  relative 
— as  a  relative  change  of  place  of  two  points — the  an- 
swer is  very  different.  If,  on  this  assumption,  we  sup- 
pose all  other  celestial  bodies  to  be  annihilated,  nothing 
remains  but  the  material  points  of  which  the  star  in 
question  itself  consists.  But,  then,  these  points  do  not 
change  their  relative  positions,  and  are  therefore  at  rest. 
It  follows  that  the  star  must  be  at  rest  at  the  moment 
when  the  annihilation  of  the  other  bodies  takes  place, 
and  therefore  must  assume  the  spherical  form  taken  by 
all  bodies  in  a  state  of  rest.  A  contradiction  so  intoler- 
able can  be  avoided  only  by  abandoning  the  assumption 
of  the  relativity  of  motion,  and  conceiving  motion  as 
absolute,  so  that  thus  we  are  again  led  to  the  principle 
of  the  body  Alpha." 

Now,  what  answer  can  be  made  to  this  reasoning  of 
Professor  Neumann  2  None,  if  we  grant  the  admissibil- 


200  CONCEPTS  OF  MODERN  PHYSICS. 

ity  of  the  hypothesis  of  the  annihilation  of  all  bodies  in 
space  but  one,  and  the  admissibility  of  the  further  as- 
sumption that  an  absolutely  rigid  body  with  an  abso- 
lutely fixed  place  in  the  universe  is  possible.  But  such 
a  concession  is  forbidden  by  the  universal  principle  of 
relativity.  In  the  first  place,  the  annihilation  of  all  bod- 
ies but  one  would  not  only  destroy  the  motion  of  this 
one  remaining  body  and  bring  it  to  rest,  as  Professor 
Neumann  sees,  but  would  also  destroy  its  very  existence 
and  bring  it  to  naught,  as  he  does  not  see.  A  body  can 
not  survive  the  system  of  relations  in  which  alone  it  has 
its  being ;  its  presence  or  position  in  space  is  no  more 
possible  without  reference  to  other  bodies  than  its  change 
of  position  or  presence  is  possible  without  such  refer- 
ence. As  has  been  abundantly  shown,  all  properties  of 
a  body  which  constitute  the  elements  of  its  distinguish- 
able presence  in  space  are  in  their  nature  relations  and 
imply  terms  beyond  the  body  itself. 

In  the  second  place  the  absolute  fixity  in  space  at- 
tributed to  the  bo'dy  Alpha  is  impossible  under  the 
known  conditions  of  reality.  The  fixity  of  a  point  in 
space  involves  the  permanence  of  its  distances  from  at 
least  four  other  fixed  points  not  in  the  same  plane. 
But  the  fixity  of  these  several  points  again  depends  on 
the  constancy  of  their  distances  from  other  fixed  points, 
and  so  on  ad  injmitum.  In  short,  the  fixity  of  position 
of  any  body  in  space  is  possible  only  on  the  supposition 
of  the  absolute  finitude  of  the  universe ;  and  this  leads 
to  the  theory  of  the  essential  curvature  of  space,  and 
the  other  theories  of  modern  transcendental  geometry, 
which  will  be  discussed  hereafter. 

There  is  but  one  issue  from  the  perplexities  of 
Euler,  and  that  is  through  the  proposition  that  the 
reality  of  rest  and  motion,  far  from  presupposing  that 


FOURTH  METAPHYSICAL  ERROR.  201 

they  are  absolute,  depends  upon  their  relativity.  The 
source  of  these  perplexities  is  readily  discovered.  It  is 
to  be  found  in  the  old  metaphysical  doctrine,  that  the 
Real  is  not  only  distinct  from,  but  the  exact  opposite 
of,  the  Phenomenal.  Phenomenalities  are  the  deliver- 
ances of  sense ;  and  these  are  said  to  be  contradictory 
of  each  other,  and  therefore  delusive.  E"ow,  the  truth 
is  that  there  is  no  physical  reality  which  is  not  phe- 
nomenal. The  only  test  of  physical  reality  is  sensible 
experience.  And  the  assertion,  that  the  testimony  of 
the  senses  is  delusive,  in  the  sense  in  which  this  asser- 
tion is  made  by  the  metaphysicians,  is  groundless.  The 
testimony  of  the  senses  is  conflicting  only  because  the 
momentary  deliverance  of  each  sense  is  fragmentary 
and  requires  control  and  rectification,  either  by  other 
deliverances  of  the  same  sense,  or  by  the  deliverances 
of  the  other  senses.  When  the  traveler  in  the  desert 
sees  before  him  a  lake  which  continually  recedes  and 
finally  disappears,  proving  to  be  the  effect  of  mirage^ 
it  is  said  that  he  is  deceived  by  his  senses,  inasmuch 
as  the  supposed  body  of  water  was  a  mere  appearance 
without  reality.  But  the  senses  were  not  deceptive. 
The  lake  was  as  real  as  the  image.  The  deception  lay 
in  the  erroneous  inferences  of  the  traveler,  who  did  not 
take  into  account  all  the  facts,  forgetting  (or  being 
ignorant  of)  the  refraction  of  the  rays  proceeding  from 
the  real  object,  whereby  their  direction  and  the  appar- 
ent position  of  the  object  were  changed.  The  true 
distinction  between  the  Apparent  and  the  Real  is  that 
the  former  is  a  partial  deliverance  of  sense  which  is 
mistaken  for  the  whole  deliverance.  The  deception 
or  illusion  results  from  the  circumstance  that  the 
senses  are  not  properly  and  exhaustively  interrogated 
and  that  their  whole  story  is  not  heard. 


202 


CONCEPTS  OF  MODERN  PHYSICS. 


The  coercive  power  of  the  prevailing  ontological 
notions  of  Euler's  time  over  the  clear  intellect  of  the 
great  mathematician  is  most  strikingly  exhibited  in  his 
statement  that  without  the  assumption  of  absolute  space 
and  motion  there  could  be  no  laws  of  motion,  so  that 
all  the  phenomena  of  physical  action  would  become 
uncertain  and  indeterminable.     If  this  argument  were 
well  founded,  the  same  consequence  would  follow,  a 
fortiori,  from  his  repeated  admissions  in  the  first  chap- 
ter of  his  book,  to  the  effect  that  we  have  no  actual 
knowledge  of  rest  and  motion,   except  that   derived 
from  bodies  at  rest  or  in  motion  in  reference  to  other 
bodies.    Euler's  proposition  can  have  no  other  meaning 
than  this,  that  the  laws  of  motion  can  not  be  established 
or  verified  unless  we  know  its  absolute  direction  and 
its  absolute  rate.    But  such  knowledge  is  by  his  own 
showing  unattainable.     It  follows,  therefore,  that  the 
establishment  and  verification  of  the  laws  of  motion  are 
impossible.     And  yet   no  one  knew  better  than  Euler 
himself  that  all  experimental  ascertainment  and  verifi- 
cation of  dynamical  laws,  like  all  acts  of  cognition,  de- 
pend upon  the  insulation  of  phenomena  ;  that  they  can 
be  effected  only  by  disentangling  the  effects  of  certain 
forces  from  the  effects  of  other  forces  (determinable 
aliunde,  i.  e.,  by  their  other  effects)  with  which  they 
are  complicated — a  proceeding  which,  in  many  cases,  is 
facilitated  by  the  circumstance  that  these  latter  effects 
are  inappreciably  small.     Surely  the  verification  of  the 
law  of  inertia  by  the  inhabitants  of  our  planet  does  not 
depend  upon  their  knowledge,  at  any  moment,  of  the 
exact  rate  of  its  angular  velocity  of  motion  round  the 
sun!    And  the  validity  of  the  Newtonian  theory   of 
celestial  motion  is  not  to  be  drawn  in  question  because 
its  author  suggests  that  the  center  of  gravity  of  our 


FOURTH  METAPHYSICAL  ERROR.         ^      203 

solar  system  moves  in  some  elliptic  orbit  whose  ele- 
ments are  not  only  unknown,  but  will  probably  never 
be  discovered  !  As  well  might  it  be  contended  that  the 
mathematical  theorems  respecting  the  properties  of  the 
ellipse  are  of  doubtful  validity,  since  no  such  curve  is 
accurately  described  by  any  celestial  body  or  can  be 
exactly  traced  by  a  human  hand  ! 

Although  in  particular  operations  of  thought  we 
may  be  constrained,  for  the  moment,  to  treat  the  Com- 
plex as  simple,  the  Variable  as  constant,  the  Transitory 
as  permanent,  and  thus  in  a  sense  to  view  phenomena 
" sub  quadam  specie  ahsoluti"  *  nevertheless  there  is 
no  truth  in  the  old  ontological  maxirn  that  the  true  na- 
ture of  things  can  be  discovered  only  by  divesting  them 
of  their  relations — that  to  be  truly  known  they  must 
be  known  as  they  are  in  themselves,  in  their  absolute 
essence.  Such  knowledge  is  impossible,  all  cognition 
being  founded  upon  a  recognition  of  relations;  and 
this  impossibility  nowhere  stands  out  in  stronger  relief 
than  in  the  exposition,  by  Newton  and  Euler,  of  the 
reality  of  rest  and  motion  under  the  conditions  of  their 
determin  ability. 

It  follows,  of  course,  from  the  essential  relativity 
of  rest  and  motion,  that  the  old  ontological  disjunction 
between  them  falls,  and  that  in  a  double  sense  rest  dif- 
fers from  motion,  in  the  language  of  Euler,  "  as  one 
motion  differs  from  another,"  f  or,  as  modern  mathe- 
maticians and  physicists  express  it,  that  "  rest  is  but  a 
special  case  of  motion."  j  And  it  follows,  furthermore, 

*  "  De  natur^,  rationis  est  res  sub  quadam  aeternitatis  specie  perci- 
pere."  Spinoza,  Eth.,  Pars,  ii,  Prop,  xliv,  Coroll.  2. 

f  u  Neque  motus  a  quiete  aliter  differt,  atque  alius  motus  ab  alio." 
Theoria  motus,  etc.,  p.  8. 

\  "  Die  Rube  ist  nur  ein  besonderer  Fall  der  Bewegung."  Kircbhoff, 
Vorlesungen  ueber  math.  Physik,  p.  32. 


204:  CONCEPTS  OF  MODERN  PHYSICS. 

that  rest  is  not  the  logically  or  cosmologically  ^i 
of  material  existence — that  it  is  not  the  natural  and 
original  state  of  '  the  universe  which  requires  no  ex- 
planation while  its  motion,  or  that  of  its  parts,  is  to 
be  accounted  for.  What  requires,  and  is  susceptible 
of,  explanation  is  always  a  change  from  a  given  state  of 
relative  rest  or  motion  of  a  finite  material  system ;  and 
the  explanation  always  consists  in  the  exhibition  of  an 
equivalent  change  in  another  material  system.  The 
question  respecting  the  origin  of  motion  in  the  uni- 
verse as  a  whole,  therefore,  admits  of  no  answer,  be- 
cause it  is  a  question  without  intelligible  meaning. 
*v  The  same  considerations  which  evince  the  relativity 
of  motion  also  attest  the  relativity  of  its  conceptual 
elements,  space  and  time.  As  to  space,  this  is  at  once 
apparent.  And  of  time.  "  the  great  independent  vari- 
able "  whose  supposed  constant  flow  is  said  to  be  the 
ultimate  measure  of  all  things,  it  is  sufficient  to  observe 
that  it  is  itself  measured  by  the  recurrence  of  certain 
relative  positions  of  objects  or  points  in  space,  and  that 
the  periods  of  this  recurrence  are  variable,  depending 
upon  variable  physical  conditions.  This  is  as  true  of  the 
data  of  our  modern  time-keepers,  the  clock  and  chro- 
nometer, as  of  those  of  the  clepsydra  and  hour-glass 
of  the  ancients,  all  of  which  are  subject  to  variations  of 
friction,  temperature,  changes  in  the  intensity  of  gravi- 
tation, according  to  the  latitude  of  the  places  of  ob- 
servation, and  so  on.  And  it  is  equally  true  of  the 
records  of  the  great  celestial  time-keepers,  the  sun  and 
the  stars.  After  we  have  reduced  our  apparent  solar 
day  to  the  mean  solar  day,  and  this,  again,  to  the  side- 
real day,  we  find  that  the  interval  between  any  two 
transits  of  the  equinoctial  points  is  not  constant,  but 
becomes  irregular  in  consequence  of  nutation,  of  the 


FOURTH  METAPHYSICAL  ERROR.         205 

precession  of  the  equinoxes,  and  of  numerous  other 
secular  perturbations  and  variations  due  to  the  mutual 
attraction  of  the  heavenly  bodies.  The  constancy  of 
the  efflux  of  time,  like  that  of  the  spatial  positions 
which  serve  as  the  basis  for  our  determination  of  the 
rates  and  amounts  of  physical  motion,  is  purely  concept- 
ual, ft 

The  relativity  of  mass  has  repeatedly  been  adverted 
to  in  the  preceding  chapters:  It  has  been  shown  that 
the  measure  of  mass  is  the  reciprocal  of  the  amount  of 
acceleration  produced  in  a  body  by  a  given  force,  while 
force,  in  turn,  is  measured  by  the  acceleration  produced 
in  a  given  mass.  It  is  readily  seen  that  the  concept 
mass  might  be  expanded,  so  as  to  assign  the  measure 
of  mass,  not  to  mechanical  motion  alone,  but  to  physi- 
cal action  generally,  including  heat  and  chemical  affin- 
ity. This  would  lead  to  an  equivalence  of  masses  dif- 
fering with  the  nature  of  the  agency  selected  as  the 
basis  of  the  comparison.  Thermally  equivalent  masses 
would  be  the  reciprocals  of  the  specific  heats  of  masses 
as  now  determined ;  and  chemically  equivalent  masses 
would  be  the  atomic  weights,  so  called.  It  is  important 
to  note  that  the  determination  of  masses  on  the  basis  of 
gravitation,  in  preference  to  their  valuation  on  the  basis 
of  thermal,  chemical  or  other  physical  action,  is  a  mere 
matter  of  convenience,  and  is  not  in  any  proper  sense 
founded  on  the  nature  of  things. 

But,  apart  from  this,  and  looking  to  the  ordinary 
method  of  determining  the  mass  of  a  body  by  its 
weight,  the  relativity  of  mass  is  equally  manifest.  The 
weight  of  a  body  is  a  function,  not  of  its  own  mass 
alone,  but  also  of  that  of  the  body  or  bodies  by  which 
it  is  attracted,  and  of  the  distance  between  them.  A 
body  whose  weight,  as  ascertained  by  the  spring-bal- 


206  CONCEPTS  OF  MODERN  PHYSICS. 

ance  or  pendulum,  is  a  pound  on  the  surface  of  the 
earth,  would  weigh  but  two  ounces  on  the  moon,  less 
than  one  fourth  of  an  ounce  on  several  of  the  smaller 
planets,  about  six  ounces  on  Mars,  two  and  one  half 
pounds  on  Jupiter,  and  more  than  twenty-seven  pounds 
on  the  sun.  And  while  the  fall  of  bodies,  in  vacuo, 
near  the  surface  of  the  earth  amounts  to  about  sixteen 
feet  (more  or  less,  according  to  the  latitude)  during 
the  first  second,  their  corresponding  fall  near  the  sur- 
face of  the  sun  is  more  than  four  hundred  and  thirty- 
five  feet. 

The  thoughtlessness  with  which  it  is  assumed  by 
some  of  the  most  eminent  physicists  that  matter  is 
composed  of  particles  which  have  an  absolute  primor- 
dial weight  persisting  in  all  positions  and  under  all  cir- 
cumstances, is  .  one  of  the  most  remarkable  facts  in  the 
history  of  science.  "The  absolute  weight  of  atoms," 
says  Professor  Redtenbacher,*  "  is  unknown  "  —  his 
meaning  being,  as  is  evident  from  the  context,  and  from 
the  whole  tenor  of  his  discussion,  that  our  ignorance 
of  this  absolute  weight  is  due  solely  to  the  practical 
impossibility  of  insulating  an  atom,  and  of  contriving 
instruments  delicate  enough  to  weigh  it. 

There  is  nothing  absolute  or  unconditioned  in  the 
world  of  objective  reality.  As  there  is  no  absolute 
standard  of  quality,  so  there  is  no  absolute  measure  of 
duration,  nor  is  there  an  absolute  system  of  coordinates 
in  space  to  which  the  positions  of  bodies  and  their 
changes  can  be  referred.  A  physical  ens  per  se  and  a 
physical  constant  are  alike  impossible,  for  all  physical 
existence  resolves  itself  into  action  and  reaction,  and 
action  imports  change. 

*  Dynamidensystcm  (Mannheim,  Bassermann,  1857),  p.  14. 


CHAPTER  XIII. 

THE   THEORY  OF    THE   ABSOLUTE   FINITUDE    OF   THE   WORLD 

AND     OF     SPACE. THE   ASSUMPTION     OF    AN    ABSOLUTE 

MAXIMUM  OF  MATERIAL  EXISTENCE  AS  A  NECESSARY 
COMPLEMENT  TO  THE  ASSUMPTION  OF  THE  ATOM  AS 
ITS  ABSOLUTE  MINIMUM. — ONTOLOGY  IN  MATHEMAT- 
ICS.  THE  REIFICATION  OF  SPACE. MODERN  TRANS- 
CENDENTAL GEOMETRY. NON-HOMALOIDAL  (SPHERICAL 

AND  PSEUDO-SPHERICAL)  SPACE. 

IT  was  shown  in  the  last  chapter,  that  the  theory, 
according  to  which  space  and  motion  are  real  only  on 
condition  of  being  absolute,  involved  the  assumption  of 
the  existence  of  an  absolutely  fixed  point  of  reference  in 
space,  and  that  this  again  of  necessity  led  to  the  doctrine 
of  the  absolute  finitude  of  the  universe.  Although 
the  connection  between  this  doctrine  and  the  prevalent 
ontological  theorems  respecting  space  and  motion  has 
not  hitherto,  so  far  as  I  am  aware,  been  pointed  out, 
the  doctrine  itself  has  been  variously  suggested  in  the 
interests  of  cosmological  speculations  founded  upon  the 
atomo-mechanical  theory,  as  a  means  of  escape  from 
certain  inevitable  consequences  of  this  theory  with  which 
those  speculations  are  found  at  last  to  conflict.  And 
it  has  recently  been  urged  by  eminent  mathematicians 
upon  considerations  respecting  the  true  nature  of  space 
and  the  real  character  of  space-relations. 

It  is  readily  seen  that  the  assertion  of  the  absolute 


208  CONCEPTS  OF  MODERN  PHYSICS. 

finitude  of  the  material  universe  is  a  logically  integral 
part  of  the  general  assertion  that  whatever  is  real  is 
absolute,  and  that  the  assumption  of  an  absolute  maxi- 
mum of  material  existence  is  a  necessary  complement 
of  the  assumption  of  its  absolute  minimum,  the  atom. 
The  first  explicit  announcement  of  a  scientific  belief  in 
this  maximum  appears  to  have  been  made  by  C.  F. 
Gauss,  in  one  of  his  letters  to  Schumacher,*  in  which 
he  discusses  the  attempts  of  his  Transylvanian  friend 
Bolyai  and  of  the  Russian  geometer  Lobatschewsky  to 
found  a  geometrical  system  which  should  be  indepen- 
dent of  the  Euclidean  axioms  in  regard  to  parallels.  The 
hints  thrown  out  by  Gauss  in  the  letters  just  referred 
to,  as  well  as  in  various  parts  of  his  other  writings,! 
have,  within  the  last  twenty  years,  been  fruitful  of  a 
discussion  respecting  the  nature  of  space,  the  founda- 
tions of  geometry  and  the  origin  and  import  of  geomet- 
rical axioms,  which  has  already  produced  an  extensive 
and  rapidly  increasing  literature.^:  The  first  effective 

*  Gauss,  Briefwechsel  mit  Schumacher,  vol.  ii,  pp.  268-2Y1. 

f  Cf.  "  Disquisitiones  generates  circa  seriem  infinitam,"  etc.  (Comm. 
recent.  Soc.  Gott.,  ii,  1811-'13);  "Theoria  residuorum  biquadraticorum 
Commentatio  secunda  (ib.,  vii,  1828-'32).  To  those  who  are  familiar  with 
Herbart's  theory  that  our  idea  of  spatial  extension  is  a  psychological 
elaborate  of  qualitative  data,  i.  e.,  of  sensations  which  are  in  themselves 
without  extension,  it  will  not  appear  improbable  that  Gauss's  mathe- 
matical transcendentalism  was  to  some  extent  due  to  the  speculations  of 
his  colleague  in  the  philosophical  faculty  of  Goettingen,  although  Gauss 
habitually  professed  great  contempt  for  the  Herbartian  system — just  as 
Descartes  was  influenced  by  the  teachings  of  his  antagonist  Gassendi. 
The  connection  of  Gauss's  metageometrical  or  (to  use  the  expression  of 
Lobatschewsky)  pangeometrical  views  with  his  investigations  respecting 
the  geometrical  interpretation  of  imaginary  quantities  and  the  theory  of 
"  complex  numbers  "  is  apparent. 

|  Cf.  Halstead,  Bibliography  of  Hyper-Space  and  non-Euclidean  Ge- 
ometry. American  Journal  of  Mathematics,  vol.  i,  pp.  261  seq.  and 
384  seq. ;  ib.  vol.  ii,  p.  65  seq. 


TRANSCENDENTAL  GEOMETRY.  209 

impulse  to  this  new  departure  in  the  walks  of  mathe- 
matical theory  was  given  by  Riemann  in  a  remarkable 
dissertation  *  read  before  the  philosophical  faculty  of 
Goettingen,  June  10,  1854  (published  by  Dedekind  in 
1866,  after  Riemann's  death),  and  by  Helmholtz  in  an 
equally  remarkable  essay  f  published  two  years  later. 
These  publications  have  since  been  followed  by  numer- 
ous articles,  pamphlets  and  books  expository  of  the 
doctrines  thus  advanced,  and,  as  was  to  be  expected, 
there  has  been  no  lack  of  writings  in  which  these  doc- 
trines have  met  with  criticism  and  denial. 

The  articles  of  the  new  geometrical  faith  are  cer- 
tainly startling.  Among  them  are  propositions  such  as 
these  :  that  our  ordinary  "  Euclidean  "  tridimensional 
and  "  homaloidal "  (flat)  space  is  but  one  of  several  pos- 
sible forms  of  space ;  that  the  preeminence  of  this  Eu- 
clidean space  among  other  forms  of  space  can  be  main- 
tained upon  empirical  grounds  alone,  and,  in  the  sense 
of  the  logical  and  psychological  tenets  of  the  sensation- 
alist school,  depends  solely  upon  the  accidents  of  no- 
tional association,  which  may  be  (and,  in  the  opinion  of 
some  enthusiastic  advocates  of  the  new  doctrines,  have 
been)  overthrown  by  the  discovery  that  the  existence  of 
additional  dimensions  is  a  necessary  inference  from  cer- 
tain facts  of  experience  which  can  not  otherwise  be  ex- 
plained— just  as  the  third  dimension  of  space  is  said  to 
be,  not  directly  perceived,  but  simply  inferred  from 
familiar  facts  of  visual  or  tactual  experience  for  whose 
explanation  the  third  dimension  is  an  indispensable 

*  Ueber  die  Hypothesen  welche  der  Geometric  zu  Grunde  liegen  (Ab- 
handlungen  der  Kgl.  Gcsellschaft  der  Wissenschaften  zu  Goettingen, 
vol.  xiii,  p.  133  seq.). 

f  Ueber  die  Thatsachen  die  der  Geometrie  zu  Grunde  liegen  (Nach- 
richten  der  Kgl.  Gesellschaft  der  Wissenschaften  zu  Goettingen,  1865, 
June  3). 


210  CONCEPTS  OF  MODERN  PHYSICS. 

postulate ;  that  true  and  real  space,  therefore,  has,  or  at 
least,  for  aught  we  know,  may  have,  not  three  but  four 
or  even  a  greater  number  of  dimensions ;  that  the  space 
in  which  we  move  is,  or  may  be,  not  homaloidal  or  flat, 
but  essentially  non  -  homaloidal,  curved,  spherical  or 
pseudo-spherical,  so  that  every  line,  which  we  have 
hitherto  regarded  as  straight,  may  upon  sufficient  pro- 
longation prove  to  be  a  closed  curve ;  that,  by  reason  of 
the  inherent  and  essential  curvature  of  space,  the  uni- 
verse, though  unlimited,  may  be,  and  probably  is,  not 
infinite,  but  finite ;  that  on  the  supposition  of  the  pseu- 
do-spherical character  of  space,  a  whole  pencil  of 
"  shortest  lines  "  may  be  drawn  through  the  same  point, 
all  which  are  parallel  to  a  given  other  "  shortest  line  " 
in  the  sense  that  they  will  never  intersect  with  it,  how- 
ever far  produced  ;  that  not  only  the  measure  of  curva- 
ture of  space,  as  well  as  the  number  of  its  dimensions, 
may  be,  and  probably  is,  different  in  different  spatial 
regions,  so  that  no  valid  inference  can  be  drawn,  from 
our  experiences  in  the  regions  in  which  we  happen  to 
dwell,  as  to  the  curvature  or  the  dimensions  of  space 
immeasurably  distant  or  immeasurably  small,  but  that 
in  any  given  region  both  the  curvature  of  space  and 
the  degree  or  number  of  its  dimensions  may  be,  and 
probably  is,  undergoing  a  gradual  transformation,  and 
so  on.* 

*  The  more  cautious  pangeometers  have  of  late  evinced  a  disposition 
to  stigmatize  some  of  the  doctrines  above  enumerated,  particularly  those 
relating  to  the  increase  in  the  number  of  spatial  dimensions  and  to  the  local 
differences  and  changes  in  the  constitution  of  space,  as  inventions  of 
their  enemies  or  as  extravagances  of  persons  who  are  carried  away  by 
their  enthusiasm.  I  may  be  pardoned,  therefore,  for  citing  a  passage 
from  a  lecture  of  Professor  P.  G.  Tait  (who  is  certainly  ready  enough,  as 
the  book  I  quote  from  shows,  to  insist  on  sobriety  in  physics  and  math- 
ematics at  least,  whatever  may,  in  his  opinion,  be  the  appropriate  frame 


TRANSCENDENTAL  GEOMETRY.  211 

However  dissonant  from  the  teachings  of  our  fa- 
miliar experience  these  propositions  seem,  it  is  claimed 
that  they  are  by  no  means  without  empirical  warrant. 

of  mind  for  surveying  the  "  Unseen  Universe ") :  "  The  properties  of 
space,"  says  Tait,  "  involving  (we  know  not  why)  the  essential  element 
of  three  dimensions,  have  recently  been  subjected  to  a  careful  scrutiny 
by  mathematicians  of  the  highest  order,  such  as  Riemann  and  Helmholtz ; 
and  the  result  of  their  inquiries  leaves  it  as  yet  undecided  whether  space 
may  or  may  not  have  precisely  the  same  properties  throughout  the  universe. 
To  obtain  an  idea  of  what  is  meant  by  such  a  statement,  consider  that  in 
crumpling  a  leaf  of  paper,  which  may  be  taken  as  representing  space  of 
two  dimensions,  we  may  have  some  portions  of  it  plane,  and  other  por- 
tions more  or  less  cylindrically  or  conically  curved.  But  an  inhabitant 
of  such  a  sheet,  though  living  in  space  of  two  dimensions  only,  and  there- 
fore, we  might  say  beforehand,  incapable  of  appreciating  the  third  di- 
mension, would  certainly  feel  some  difference  of  sensations  in  passing 
from  portions  of  his  space  which  were  less  to  other  portions  which  were 
more  curved.  So  it  is  possible  that,  in  the  rapid  march  of  the  solar  system 
through  space,  we  may  be  gradually  passing  to  regions  in  which  space  has 
not  precisely  the  same  properties  as  we  find  here — where  it  may  have  some- 
thing in  three  dimensions  analogous  to  curvature  in  two  dimensions — some- 
thing, in  fact,  which  will  necessarily  imply  a  fourth-dimension  change  of 
form  in  portions  of  matter  in  order  that  they  may  adapt  themselves  to  their 
new  locality."  P.  G.  Tait,  On  Some  Recent  Advances  in  Physical  Science, 
p.  6.  In  keeping  with  this  passage  is  a  note  of  the  distinguished  mathe- 
matician, Professor  J.  J.  Sylvester,  to  his  opening  address  to  the  Mathe- 
matical and  Physical  Section  of  the  British  Association  at  Exeter,  in 
1869,  as  follows:  "It  is  well  known,  to  those  who  have  gone  into  these 
views,  that  the  laws  of  motion  accepted  as  a  fact  suffice  to  prove  in  a 
general  way  that  the  space  we  live  in  is  a  flat  or  level  space  (a  *  homa- 
loid '),  our  existence  therein  being  assimilable  to  the  life  of  a  bookworm 
in  the  flat  space  ;  but  what  if  Hue  page  should  be  undergoing  a  process  of 
gradual  bending  into  a  curved  form  ?  Mr.  W.  K.  Clifford  has  indulged 
in  some  remarkable  speculations  as  to  the  possibility  of  our  being  able 
to  infer,  from  certain  unexplained  phenomena  of  light  and  magnetism, 
the  fact  of  our  level  space  of  three  dimensions  being  in  the  act  of  under- 
going in  space  of  four  dimensions  (space  as  inconceivable  to  us  as  our 
space  to  our  supposititious  bookworm)  a  distortion  analogous  to  the  rum- 
pling of  the  page.  I  know  there  are  many  who,  like  my  honored  and 
deeply  lamented  friend,  the  late  eminent  Professor  Donkin,  regard  the 
alleged  notion  of  generalized  space  as  only  a  disguised  form  of  algebrai- 


212  CONCEPTS  OF  MODERN  PHYSICS. 

It  is  insisted  that  there  are  numerous  optic,  magnetic 
and  other  physical  phenomena  of  which  they  yield  the 
only  sufficient  explanation.  Moreover,  it  is  said  that 
they  alone  afford  a  clew  to  the  mysteries  of  modern 
spiritism,  enabling  us  to  bring  within  the  chain  of  nat- 
ural causation  certain  magical  performances  which  we 
should  otherwise  be  constrained  to  relegate  to  the  regions 
of  the  Supernatural.  In  the  first  article  of  the  first  num- 
ber of  the  American  Journal  of  Mathematics,  Professor 
Simon  Newcomb  demonstrates  analytically  that,  "  if  a 
fourth  dimension  were  added  to  space,  a  closed  material 
surface  (or  shell)  could  be  turned  inside  out  by  simple 
flexure  without  either  stretching  or  tearing,"  Felix 
Klein  having  shown,  some  time  before,  that  knots  can 
not  exist  in  a  four-dimensional  space.  Accordingly, 
Professor  Zoellner  accounts  for  the  well-known  feats  of 
the  American  "  medium  "  Slade  on  the  principle  of  the 
fourth  dimension — one  of  these  feats,  however,  strange- 
ly enough,  consisting  in  the  production  of  real  trefoil 
knots  in  a  rope  the  ends  of  which  were  sealed  together 
and  held  in  Zoellner's  hands.  And,  finally,  it  is  as- 
serted that  the  theorems  of  Lobatschewsky,  Riemann, 

cal  formulization ;  but  the  same  might  be  said  with  equal  truth  of  our 
notion  of  infinity  in  algebra,  or  of  impossible  lines,  or  lines  making  a 
zero  angle  in  geometry,  the  utility  of  dealing  with  which  as  positive  sub- 
stantiated notions  no  one  will  be  found  to  dispute.  Dr.  Salmon,  in  his 
extensions  of  Chasles's  theory  of  characteristics  to  surfaces,  Mr.  Clifford 
in  a  question  of  probability,  and  myself  in  my  theory  of  partitions,  and 
also  in  my  paper  on  Barycentric  Projection,  in  the  Philosophical  Magazine, 
have  all  felt  and  given  evidence  of  the  practical  utility  of  handling  space 
of  four  dimensions  as  if  it  were  conceivable  space.  Moreover,  it  should 
be  borne  in  mind,  that  every  perspective  representation  of  figured  space 
of  four  dimensions  is  a  figure  in  real  space,  and  that  the  properties  of 
figures  admit  of  being  studied,  to  a  great  extent,  if  not  completely,  in 
their  perspective  representations."  Nature,  vol.  i,  p.  237  seq.  The  ital- 
ics in  the  above  passages  are  mine. 


TRANSCENDENTAL  GEOMETRY.  213 

Helmholtz  and  Beltrami,*  are  the  only  true  basis  of  a 
proper  and  exhaustive  theory  of  parallelism.  In  the 
fullness  of  their  faith  in  the  impregnability  of  these 
positions,  the  votaries  at  the  shrine  of  geometrical  tran- 
scendentalism make  bold  to  announce  that,  with  the 
appearance  of  Lobatschewsky' s  "  Geometrical  Investiga- 
tions," f  a  new  era  nas  dawned  upon  the  mathematical 
world,  and  that  in  the  daylight  of  this  era  the  whole 
body  of  geometrical  truths  will  be  reduced  to  simplicity 
and  order  in  a  way  analogous  to  that  in  which  the  the- 
ory of  celestial  motions  was  simplified  and  cleared  up 
by  the  great  thought  of  Copernicus.  "  What  Vesalius 
was  to  Galen,"  exclaims  Professor  Clifford,:):  "what 
Copernicus  was  to  Ptolemy,  that  was  Lobatschewsky  to 
Euclid." 

The  debate  between  the  disciples  of  the  new  tran-1 
scendental  or  pangeometrical  school  and  the  adherents 
of  the  old  geometrical  faith  presents  one  feature  which 
can  not  fail  to  strike  the  ordinary  observer  with  some 
amazement.  The  disciples  of  the  new  school  take  their 
stand  firmly  upon  empirical  ground ;  their  very  first 
proposition  is  that  all  geometrical  truths  are  of  empiri- 
cal origin,  and  that  all  we  know  of  space  and  its  prop- 
erties is  what  we  are  taught  by  sensible  experience. 
This  proposition  and  the  consequent  denial  of  the 

*  An  Italian  mathematician  who  has  investigated  the  properties  of 
pseudo-spherical  surfaces,  which  are  distinguished  from  other  surfaces  of 
constant  curvature  by  the  fact  that  they  admit  of  a  sort  of  parallelism, 
in  the  transcendental  sense,  between  their  "  straightest  lines."  A  refer- 
ence to  Beltrami's  writings  and  a  brief  exposition  of  their  contents  may- 
be found  in  Helmholtz's  essay  on  "  The  Origin  and  Meaning  of  Geometri- 
cal Axioms,"  Mind,  vol.  i,  p.  306. 

f  Geometrische  Untersuchungen  zur  Theorie  der  Parallellinien,  von 
Nicolaus  Lobatschewsky.  Berlin,  Fincke'sche  Buchhandlung,  1840. 

$  Philosophy  of  the  Pure  Sciences,  W.  K.  Clifford's  Lectures  and  Es- 
says, vol.  i,  p.  297. 
10 


214:  CONCEPTS  OF  MODERN  PHYSICS. 

transcendental  origin  of  geometrical  axioms  are  empha- 
zised  by  Riemann  and  Helmholtz  alike.  And  yet,  upon 
this  foundation  they  construct  a  theory  which  lands  us 
in  the  remotest  regions  of  transcendentalism — in  the 
realms  of  a  metageonaetrical  space  in  which  all  our 
wonted  powers  of  imagination  and  conception  are  at 
fault  and  in  which  the  facts  of  every-day  experience  as 
well  as  their  natural  relations  are  wholly  out  of  sight. 
On  the  other  hand,  the  most  conspicuous  champions  of 
the  old  geometrical  creed,  in  their  defense  of  the  famil- 
iar data  of  sensible  experience  and  in  their  antagonism 
to  the  "  vagaries  "  of  transcendental  geometry,  invoke 
the  doctrine  of  the  non-empirical  or  transcendental 
origin  of  our  ideas  of  space  and  its  essential  relations. 
The  pangeometers  erect  a  transcendental  structure  on 
empirical  foundations,  while  the  ordinary  geometers 
build  a  system  conforming  to  the  data  of  experience 
upon  transcendental  grounds.  This  circumstance,  how- 
ever, strange  as  it  appears  at  first  sight,  will  hardly 
surprise  the  thoughtful  student  of  the  history  of  theo- 
ries of  cognition,  or  the  intelligent  reader  of  the  pre- 
ceding pages.  It  is  by  no  means  unusual  to  find  that 
ontological  speculations,  whether  they  appear  in  the 
guise  of  physical  or  in  that  of  metaphysical  theories, 
prove  subversive  in  the  end,  not  merely  of  the  facts  for 
whose  explanation  they  were  devised,  but  of  the  very 
supports  by  which  they  are  supposed  to  be  upheld. 

Having  indicated,  generally,  the  purport  and  scope 
of  the  transcendental  theory  of  space,  I  now  proceed  to 
the  examination  of  the  premisses  upon  which  it  rests 
and  of  the  arguments  by  which  it  is  sought  to  be  sus- 
tained. Here,  at  the  outset,  we  find  an  assumption 
which  obviously  lies  at  the  base  of  the  whole  theory : 
the  assumption  that  space  is  a  physically  real  thing — 


TRANSCENDENTAL  GEOMETRY.  215 

not  merely  an  object  of  experience,  but  an  independent 
object  of  direct  sensation  whose  properties  may  be  as- 
certained by  the  aid  of  the  ordinary  instruments  of 
physical  and  astronomical  research — whose  degree  of 
curvature,  for  instance,  is  to  be  determined  by  means 
of  the  telescope.  This  assumption  is  explicitly  stated 
by  each  of  the  three  great  expounders  of  the  theory  in 
question.  "  The  only  means  at  our  command,"  says  Lo- 
batschewsky,*  "  to  determine  the  accuracy  of  the  prop- 
ositions (calculations)  of  ordinary  geometry  consist  in 
an  appeal  to  astronomical  observations."  To  the  same 
effect  Riemann  :  f  "If  we  assume  that  bodies  exist  in- 
dependently of  their  location  in  space,  the  measure  of 
curvature  (of  space)  is  everywhere  constant ;  and  then 
it  follows  from  astronomical  measurements  that  it  is 
not  different  from  zero."  And  in  the  same  sense 
Helmholtz  :  \  "  All  systems  of  practical  mensuration 
that  have  been  used  for  the  angles  of  large  rectilinear 
triangles,  and  especially  all  systems  of  astronomical 
measurement  which  make  the  parallax  of  immeasura- 
bly distant  fixed  stars  equal  to  zero  (in  pseudo-spherical 
space  the  parallax  even  of  infinitely  distant  points 
would  be  positive),  confirm  empirically  the  axiom  of 
parallels  and  show  the  measure  of  curvature  of  our 
space  thus  far  to  be  indistinguishable  from  zero.  It 
remains,  however,  a  question,  as  Riemann  observed, 
whether  the  result  might  not  be  different  if  we  could 
use  other  than  our  limited  base-lines,  the  greatest  of 
which  is  the  major  axis  of  the  earth's  orbit" 

The  view  thus  taken  of  the  nature  of  space  and  of 

*  Geometrische  Untersucliungen,  etc.,  p.  60. 

f  Ueber  die  Hypothesen,  etc.,  Abhandl.  der  Kgl.  Gesellschaft  der 
Wissenschaften  zu  Goettingen,  vol.  xiii,  p.  148. 

\  "On  the  Origin  and  Meaning  of  Geometrical  Axioms,"  Mind,  vol.  i, 
p.  314. 


216  CONCEPTS  OF  MODERN  PHYSICS. 

the  origin  of  our  notions  concerning  it  is  obviously  in- 
dicative of  a  decided  advance  beyond  the  farthest  out- 
posts of  the  old  sensationalist  camp.  Nevertheless,  it 
is  supported  in  the  main  by  a  reference  to  the  writings 
of  a  British  thinker,  J.  S.  Mill,  who  has  been  repeated- 
ly referred  to  in  these  pages,  and  who  is  regarded,  es- 
pecially on  the  Continent,  as  the  ablest  modern  ex- 
pounder and  defender  of  the  doctrines  of  sensational- 
ism, so  far,  at  least,  as  they  bear  upon  the  subject  now 
under  consideration.*  Stated  in  brief  words,  these 
doctrines  are  that  the  idea  or  notion  of  space  is  directly 
derived  from  sensible  experience ;  that  the  properties 
of  space  are  to  be  determined  by  observation  or  ex- 
periment ;  that  the  fundamental  truths  of  geometry, 
like  all  other  truths  of  physical  science,  are  of  induc- 
tive origin  and  warrant ;  and  that  the  certainty  to  be 
attributed  to  geometrical  theorems,  though  possibly 
different  in  degree,  is  not  different  in  kind  from  that 
belonging  to  any  general  assertion  respecting  physical 
facts.  The  peculiar  tenets  of  pangeometry  being  thus 
founded,  in  great  part  at  least,  upon  the  general  sensa- 
tionalist theory,  it  will  be  useful  to  enter  upon  a  brief 
examination  of  this  theory  before  I  proceed  to  discuss 
the  pangeometrical  tenets  themselves.  For  this  pur- 

*  I  do  not  mean  to  say  that  Riemann  and  Helmholtz  themselves  di- 
rectly refer  to  Mill.  But  there  are  few  German  physicists  and  mathema- 
ticians who  have  not  been  diligent  students  of  Mill's  Logic,  particularly 
since  the  appearance  of  Schiel's  translation  and  the  extravagant  praises 
of  Liebig ;  and  this  is  quite  apparent  in  most  of  the  writings  of  the 
pangeometers.  The  interest  with  which  each  new  edition  of  Mill's  Logic 
has  been  received  by  scientific  men  everywhere  is  mainly  due,  doubtless, 
to  its  frequent  references  to  scientific  methods  and  results.  The  fact  is 
that  Mill  has,  for  a  series  of  years,  been  the  official  logician  and  meta- 
physician of  the  Continental  naturalists  and  mathematicians,  and  the  re- 
gard in  which  he  is  held  by  contemporary  men  of  science  is  not  unlike 
that  which  Aristotle  enjoyed  among  the  early  mediaeval  scholastics. 


TRANSCENDENTAL  GEOMETRY.  217 

pose  I  select  an  exposition  of  the  theory  in  the  book 
above  referred  to,  the  System  of  Logic  of  J.  S.  Mill, 
in  which  the  fifth  chapter  of  the  second  book  "On 
Demonstration  and  Necessary  Truth  "  contains  an  elab- 
orate statement  of  the  author's  views  on  the  basis  and 
methods  of  geometrical  science. 

"  The  foundation  of  all  sciences,  even  deductive  or 
demonstrative  sciences,"  says  Mill  *  "  is  Induction ; 
every  step  in  the  ratiocination  of  geometry  is  an  act  of 
induction.  .  .  .  The  character  of  necessity  ascribed  to 
the  truths  of  mathematics,  and  even  (with  some  reser- 
vations to  be  hereafter  made)  the  peculiar  certainty  at- 
tributed to  them  is  an  illusion;  in  order  to  sustain 
which  it  is  necessary  to  suppose  that  those  truths  relate 
to,  and  express,  the  properties  of  purely  imaginary  ob- 
jects. It  is  acknowledged  that  the  conclusions  of  ge- 
ometry are  deduced,  partly  at  least,  from  the  so-called 
Definitions,  and  that  those  definitions  are  assumed  to 
be  correct  representations,  as  far  as  they  go,  of  the  ob- 
jects with  which  geometry  is  conversant.  Now,  we  have 
pointed  out  that,  from  a  definition  as  such,  no  proposi- 
tion, unless  it  be  one  concerning  the  meaning  of  a  word, 
can  ever  follow ;  and  that  what  apparently  follows  from 
a  definition  follows  in  reality  from  an  implied  assump- 
tion that  there  exists  a  real  thing  conformable  thereto. 
This  assumption,  in  the  case  of  the  definitions  of  geom- 
etry, is  not  strictly  true  ;  there  exist  no  real  things  ex- 
actly conformable  to  the  definitions.  There  exist  no 
points  without  magnitude;  no  lines  without  breadth, 
nor  perfectly  straight;  no  circles  with  all  their  radii 
exactly  equal,  nor  squares  with  all  their  angles  perfectly 
right.  It  will  perhaps  be  said  that  the  assumption  does 
not  extend  to  the  actual,  but  only  to  the  possible,  exist- 

*  A  System  of  Logic  (eighth  ed.),  p.  168  seq. 


218  CONCEPTS  OF  MODERN  PHYSICS. 

ence  of  such  things.  I  answer  that,  according  to  any 
test  we  have  of  possibility,  they  are  not  even  possible. 
Their  existence,  so  far  as  we  can  form  any  judgment, 
would  seem  to  be  inconsistent  with  the  physical  consti- 
tution of  our  planet  at  least,  if  not  of  the  universe.  To 
get  rid  of  this  difficulty,  and  at  the  same  time  to  save 
the  credit  of  the  supposed  system  of  necessary  truth,  it 
is  customary  to  say  that  the  points,  lines,  circles,  and 
squares  which  are  the  subject  of  geometry  exist  in  our 
conceptions  merely,  and  are  part  of  our  minds ;  which 
minds,  by  working  on  their  own  materials,  construct  an 
a  priori  science,  the  evidence  of  which  is  purely  men- 
tal, and  has  nothing  whatever  to  do  with  outward  ex- 
perience. By  however  high  authorities  this  doctrine 
may  have  been  sanctioned,  it  appears  to  me  psychologi- 
cally incorrect.  The  points,  lines,  circles,  and  squares 
which  any  one  has  in  his  mind  are  (I  apprehend)  simply 
copies  of  the  points,  lines,  circles,  and  squares  which  he 
has  known  in  his  experience.  Our  idea  of  a  point  I 
apprehend  to  be  simply  our  idea  of  the  minimum  visi- 
fiile,  the  smallest  portion  of  surface  which  we  can  see. 
A  line,  as  defined  by  geometers,  is  wholly  inconceivable. 
We  can  reason  about  a  line  as  if  it  had  no  breadth ;  be- 
cause we  have  a  power,  when  a  perception  is  present  to 
our  senses,  or  a  conception  to  our  intellects,  of  attending 
to  a  part  only  of  that  perception  or  conception,  instead 
of  the  whole.  But  we  can  not  conceive  a  line  without 
breadth  ;  we  can  form  no  mental  picture  of  such  a 
line  ;  all  the  lines  which  we  have  in  our  minds  are  lines 
possessing  breadth.  If  any  one  doubts  this,  we  may  re- 
fer him  to  his  own  experience.  I  much  question  if  any 
one,  who  fancies  that  he  can  conceive  what  is  called  a 
mathematical  line,  thinks  so  from  the  evidence  of  his 
consciousness :  I  suspect  it  is  rather  because  he  supposes 


TRANSCENDENTAL  GEOMETRY.  219 

that,  unless  such  a  conception  were  possible,  mathematics 
could  not  exist  as  a  science :  a  supposition  which  there 
will  be  no  difficulty  in  showing  to  be  entirely  ground- 
less. 

"  Since,  then,  neither  in  nature,  nor  in  the  human 
mind,  do  there  exist  any  objects  exactly  corresponding 
to  the  definitions  of  geometry,  while  yet  that  science 
can  not  be  supposed  to  be  conversant  about  nonentities, 
nothing  remains  but  to  consider  geometry  as  conversant 
with  such  lines,  angles,  and  figures,  as  really  exist ;  and 
the  definitions,  as  they  are  called,  must  be  regarded  as 
some  of  our  first  and  most  obvious  generalizations  con- 
cerning those  natural  objects.  The  correctness  of  those 
generalizations,  as  generalizations,  is  without  a  flaw ; 
the  equality  of  all  the  radii  of  a  circle  is  true  of  all  cir- 
cles, so  far  as  it  is  true  of  any  one  :  but  it  is  not  exactly 
true  of  any  circle ;  it  is  only  nearly  true  ;  so  nearly  that 
no  error  of  any  importance  in  practice  will  be  incurred 
by  feigning  it  to  be  exactly  true.  When  we  have  occa- 
sion to  extend  these  inductions,  or  their  consequences, 
to  cases  in  which  the  error  would  be  appreciable — to 
lines  of  perceptible  breadth  or  thickness,  parallels  which 
deviate  sensibly  from  equidistance,  and  the  like — we 
correct  our  conclusions  by  combining  them  with  a  fresh 
set  of  propositions  relating  to  the  aberration  ;  just  as  we 
also  take  in  propositions  relating  to  the  physical  or 
chemical  properties  of  the  material  if  those  properties 
happen  to  introduce  any  modification  into  the  result ; 
which  they  easily  may,  even  with  respect  to  figure  and 
magnitude,  as  in  the  case,  for  instance,  of  expansion  by 
heat.  So  long,  however,  as  there  exists  no  practical 
necessity  for  attending  to  any  of  the  properties  of  the 
object  except  its  geometrical  properties,  or  any  of  the 
natural  irregularities  in  those,  it  is  convenient  to  neglect 


220  CONCEPTS  OF  MODERN  PHYSICS. 

the  consideration  of  the  other  properties  and  of  the  ir- 
regularities, and  to  reason  as  if  these  did  not  exist :  ac- 
cordingly, we  formally  announce  in  the  definitions  that 
we  intend  to  proceed  on  this  plan.  But  it  is  an  error 
to  suppose,  because  we  resolve  to  confine  our  attention 
to  a  certain  number  of  the  properties  of  an  object,  that 
we  therefore  conceive,  or  have  an  idea  of,  the  object, 
denuded  of  its  other  properties.  We  are  thinking,  all 
the  time,  of  precisely  such  objects  as  we  have  seen  and 
touched,  and  with  all  the  properties  which  naturally  be- 
long to  them  ;  but,  for  scientific  convenience,  we  feign 
them  to  be  divested  of  all  properties,  except  those  which 
are  material  to  our  purpose,  and  in  regard  to  which  we 
design  to  consider  them. 

"  The  peculiar  accuracy,  supposed  to  be  character- 
istic of  the  first  principles  of  geometry,  thus  appears  to 
be  fictitious.  The  assertions  on  which  the  reasonings 
of  the  science  are  founded  do  not,  any  more  than  in 
other  sciences,  exactly  correspond  with  the  fact;  but 
we  suppose  that  they  do  so,  for  the  sake  of  tracing  the 
consequences  which  follow  from  the  supposition.  The 
opinion  of  Dugald  Stewart  respecting  the  foundations 
of  geometry  is,  I  conceive,  substantially  correct ;  that  it 
is  built  on  hypotheses ;  that  it  owes  to  this  alone  the 
peculiar  certainty  supposed  to  distinguish  it ;  and  that 
in  any  science  whatever,  by  reasoning  from  a  set  of 
hypotheses,  we  may  obtain  a  body  of  conclusions  as  cer- 
tain as  those  of  geometry,  that  is,  as  strictly  in  accord- 
ance with  the  hypotheses,  and  as  irresistibly  compelling 
assent,  on  condition  that  those  hypotheses  are  true.',O 

I  have  quoted  this  passage,  from  Mill's  Logic,  at 
length,  not  only  because  it  is  the  most  elaborate  and 
connected  statement  of  the  sensationalist  theories  con- 
cerning the  character  of  necessary  truths  and  especially 


TRANSCENDENTAL  GEOMETRY.  ',       221 

the  truths  of  geometry,  but  also  because  this  statement 
exhibits  certain  peculiarities  that  are  worthy  of  atten- 
tion. One  of  these  peculiarities  is  the  concession  that 
the  mind  has  the  power  of  abstraction,  and  of  forming 
and  reasoning  about  generalizations  which,  "  as  generali- 
zations, are  without  a  flaw."  The  inconsistency  of  this 
admission  with  the  claim  that  "  the  points,  lines,  circles, 
and  squares  which  any  one  has  in  his  mind  are  simply 
copies  of  the  points,  lines,  circles,  and  squares  which  he 
has  known  in  his  experience,"  is  evident.  And  this 
inconsistency  has  not  escaped  the  notice  of  other  pro- 
moters of  the  experiential  or  sensationalist  doctrine,  as 
is  shown,  for  instance,  in  the  writings  of  Mr.  Buckle, 
who  does  not  hesitate  to  draw  the  true  conclusions 
(from  which  Mill  himself  appears  to  recoil)  from  Mill's 
premisses.  Buckle  not  only  boldly  asserts  that  there 
are  no  lines  without  breadth  (he  strangely  forgets  the 
thickness),  but  also  that  the  neglect  of  this  breadth  by 
the  geometrician  vitiates  all  the  results  of  geometrical 
inference,  the  only  comfort  vouchsafed  to  us  being  that 
the  error,  after  all,  is  not  very  considerable.  "  Since, 
however,"  he  says,*  "  the  breadth  of  the  faintest  line  is 
so  slight  as  to  be  incapable  of  measurement,  except  by 
an  instrument  used  under  the  microscope,  it  follows 
that  the  assumption  that  there  can  be  lines  without 
breadth  is  so  nearly  true  that  our  senses,  when  unas- 
sisted by  art,  can  not  detect  the  error.  Formerly,  and 
until  the  invention  of  the  micrometer,  in  the  seven- 
teenth century,  it  was  impossible  to  detect  it  at  all. 
Hence,  the  conclusions  of  the  geometrician  approximate 
so  closely  to  truth  that  we  are  justified  in  accepting 
them  as  true.  The  flaw  is  too  minute  to  be  perceived. 

*  History  of  Civilization  in  England,  vol.  ii,  p.  342  (Appletons'  Ameri- 
can edition). 


222  CONCEPTS  OF  MODERN  PHYSICS. 

But  that  there  is  a  flaw  appears  to  me  certain.  It  ap- 
pears certain  that,  whenever  something  is  kept  back  in 
the  premisses,  something  must  be  wanting  in  the  con- 
clusion. In  all  such  cases,  the  field  of  inquiry  has  not 
been  entirely  covered ;  and,  part  of  the  preliminary  facts 
being  suppressed,  it  must,  I  think,  be  admitted  that 
complete  truth  be  unattainable,  and  that  no  problem  in 
geometry  has  been  exhaustively  solved." 

Whether  Buckle  was  able  to  think  of  a  line  as  the 
limit  between  two  surfaces,  and  whether,  in  his  opinion, 
such  a  limit  has  breadth  (i.  e.,  is  itself  a  surface,  so  that 
we  are  driven  from  limit  to  limit,  ad  infinituni),  he 
does  not  tell  us.  Nor  does  he  say  whether  or  not,  in 
view  of  the  fact  that  the  breadth  of  a  line  depends  upon 
the  material  out  of  which  it  is  constructed,  or  upon 
which  it  is  drawn,  there  ought  to  be  a  pasteboard  geom- 
etry, a  wooden  geometry,  a  stone  geometry,  and  so  on, 
as  distinct  sciences. 

To  do  justice,  however,  to  Mill  and  the  subject 
under  discussion,  we  must  keep  before  us  Mill's  own 
statement.  Returning,  then,  to  his  exposition,  the 
question  arises  at  once :  What  does  he  mean  when  he 
says  that  none  of  the  elements  of  space  exist  in  fact  as 
they  are  considered  in  the  science  of  geometry — that, 
for  example,  there  exist  no  lines  perfectly  straight? 
The  only  possible  meaning  is  that  none  of  the  straight 
lines,  so  called,  of  which  we  have  experiential  knowl- 
edge, are  congruent  with  the  straight  lines  of  which  we 
have  other  knowledge — that  they  do  not  conform  to  the 
standard  straight  line  in  our  minds.  But  Mill  asserts 
that  "  the  lines,  etc.,  which  any  one  has  in  his  mind, 
are  simply  copies  of  the  lines  which  he  has  known  in 
his  experience."  There  is  no  standard,  therefore,  with 
which  the  lines  presented  in  experience  can  be  compared 


TRANSCENDENTAL  GEOMETRY.  „       223 

and  from  which  they  can  be  shown  to  be  divergent. 
Thus  Mill's  theory  breaks  down  with  the  very  first  fact 
which  he  brings  to  its  support.*  This  is  no  mere  cap- 
tious criticism.  It  is  a  simple  exhibition  of  the  utter 
senselessness  of  the  premisses  from  which  Mill's  con- 
clusions are  drawn.  The  whole  foundation  of  his  theory 
crumbles  the  moment  it  is  touched.  And,  upon  further 
examination,  it  is  found  that  he  entirely  mistakes  the 
significance  of  the  facts  which  he  adduces.  The  real 
import  of  Mill's  assertion  just  referred  to  is  very  differ- 
ent from  that  which  he  ascribes  to  it.  The  truth  which 
lies  at  the  bottom  of  that  assertion  is  that  we  have  no 
experiential  knowledge,  in  Mill's  sense,  of  lines,  circles, 
squares,  at  all.  We  have  experiential  knowledge  of 
so-called  straight  rods,  cords,  edges,  or  grooves,  of 
spherical  and  cubical  bodies  with  circular  or  square 
sections  or  sides;  but  our  knowledge  of  points,  lines, 
surfaces,  and  geometrical  solids  is  obtained  solely  by 
the  process  of  abstraction.  Nothing  is  clearer  and  more 
readily  demonstrable  than  that  the  elements  of  geomet- 
rical science — the  foundations  upon  which  the  science 
of  geometry  rests — can  not  have  been  obtained  by  in- 
duction, and  that,  a  fortiori,  it  can  not  be  true,  as  Mill 
contends,  that  "  every  step  in  the  ratiocinations  of 
geometry  is  an  act  of  induction.1"  Induction  is  a  cu- 

*  That  so  acute  a  thinker  as  J.  S.  Mill  was  blind  to  the  many  incon- 
sistencies and  absurdities  with  which  his  Logic  and  parts  of  his  other 
writings  abound,  is  explicable  solely  by  the  fact  that  he  took  his  theory  of 
cognition  upon  trust  as  a  sacred  inheritance  from  his  father,  who,  in  turn, 
had  derived  it  from  the  French  and  English  nominalists  and  sensational- 
ists of  the  seventeenth  and  eighteenth  centuries.  The  doctrines  of  these 
sensationalists  were  necessarily  crude,  because  they  originated  at  a  time 
when  rational  psychology  was  in  its  infancy  and  comparative  psychology 
was  not  even  thought  of ;  and  they  were  extravagant  because  they  were 
generated  by  the  antagonisms  of  an  equally  extravagant  realism. 


224  .CONCEPTS  OF  MODERN  PHYSICS. 

mulation  of  instances  in  all  of  which  the  same  element 
or  feature  is  found  along  with  other  elements  or  feat- 
ures. But  no  one  has  ever  seen  two  bodies  whose 
edges,  though  called  straight,  did  not  prove  to  be  broken 
in  different  degrees  when  examined  with  sufficient 
magnifying  power.  Experience  furnishes  no  two  in- 
stances presenting  the  feature  of  straightness  in  the 
same  degree.  Much  less  has  any  one  seen  a  great  num- 
ber of  bodies  whose  edges  were  exactly  coincident. 
The  same  thing  is,  of  course,  true,  mutatis  mutandis, 
of  points,  curves,  surfaces,  and  solids.  The  divergences 
between  their  figures  as  well  as  their  magnitudes  become 
more  apparent  in  proportion  to  the  magnifying  power 
with  which  they  are  examined.  And  their  true  figures 
are  wholly  unoliscoverable  by  any  magnifying  process 
at  our  command.  The  truth  is,  that  we  never  get  sight 
or  come  into  the  actual  presence  of  a  true  and  complete 
geometrical  fact.  It  is  simply  nonsense,  therefore,  to 
say,  with  Mill,  that  the  points,  lines,  surfaces,  solids, 
etc.,  with  which  the  science  of  geometry  deals  and 
respecting  which  it  is  able  to  draw  valid  deductions, 
are  real  (i.  e.,  physical)  and  not  imaginary  points,  lines, 
surfaces,  and  solids,  and  that  the  points,  lines,  surfaces, 
and  solids  in  our  minds  are  copies  of  them.  It  is  true 
enough  that  the  geometrical  elements  are  not  imaginary, 
because  they  have  reference  to  real  facts  ;  nor  are  they 
in  any  proper  sense  hypothetical,  as  is  contended  by 
Dugald  Stewart ;  they  are  conceptual^  the  results  of 
abstraction.  If  it  were  otherwise,  deductive  geomet- 
rical ratiocination — and,  indeed,  any  other  kind  of  rea- 
soning properly  so  called — would  be  utterly  impossible. 
All  deductive  reasoning  depends  upon  the  power  of 
abstraction.  And  this  truth  is  applicable,  not  only  to 
geometry  and  to  mathematics  generally,  but  to  all 


TRANSCENDENTAL  GEOMETRY.  225 

sciences  whatever.  This  is  so  for  two  reasons  :  In  the 
first  place,  no  physical  thing  (or  historical  event)  ever 
becomes  experientially  known  to  us  with  all  its  prop- 
erties, relations,  or  incidents ;  sensation  and  perception 
never  furnish  the  intellect  with  a  complete  fact.  And, 
in  the  second  place,  as  I  have  heretofore  shown,  in 
dealing  with  the  facts,  so  called,  obtained  at  the  hands 
of  sensible  experience,  the  intellect  is  restricted  to  cer- 
tain definite  relations  which  it  segregates  or  abstracts 
from  other  relations.  In  the  processes  of  discursive 
thought,  the  intellect  never  has  before  it  either  sensible 
objects  or  the  whole  complement  of  relations  which 
make  up  their  mental  images  or  representations,  but 
only  some  single  relation  or  class  of  relations.  It  oper- 
ates along  lines  of  abstraction,  the  final  synthesis  of 
whose  results  never  yields  anything  more  than  outlines 
of  the  objects  represented.  During  all  its  operations 
the  intellect  is  fully  aware  that  neither  any  one  link  in 
the  chain  of  abstraction  nor  the  group  of  abstract  results 
which  we  call  a  concept  (in  the  narrower  sense  of  a 
collection  of  attributes  representing  an  object  of  intui- 
tion or  sensation)  is  a  copy  or  an  exact  counterpart  of 
the  object  represented.  It  is  always  conscious  that,  to 
bring  about  true  conformity  between  concepts  or  any 
of  their  constituent  attributes  with  forms  of  objective 
reality,  the  group  of  relations  embodied  in  these  con- 
cepts would  have  to  be  supplemented  with  an  inde- 
terminate number  of  other  relations  which  have  not 
been  apprehended  and  possibly  are  insusceptible  of 
apprehension.  But  this  nowise  affects  the  validity  of 
the  intellectual  operation.  The  mathematician,  when 
he  determines  the  properties  of  a  conic  section,  knows 
full  well  that  he  will  never  meet  with  a  body  whose 
geometrical  outline  is  an  exact  exemplification  of  the 


226  CONCEPTS  OF  MODERN  PHYSICS. 

law  of  the  constancy  of  the  ratio  between  the  distances 
of  any  one  of  its  points  from  a  fixed  point  and  a  fixed 
straight  line  respectively,  and  that  there  is  in  nature  no 
trajectory  which  strictly  coincides  with  such  a  curve. 
But  this  knowledge  does  not  in  the  least  degree  disturb 
his  faith  in  the  perfect  validity  of  his  reasoning.  When 
he  comes  to  apply  the  results  of  this  reasoning  to  a  nat- 
ural fact,  he  supplements  it,  as  well  as  he  can,  with  the 
results  of  other  processes  of  reasoning  based  upon  other 
known  relations  of  the  same  fact,  and  thus  approaches 
the  fact  as  nearly  as  possible,  nothing  daunted  by  the 
ever-present  reflection  that  he  will  never  succeed  in 
coming  into  the  actual  presence  of  the  whole  fact  with 
all  its  relations. 

It  is  obvious  that  the  conformity  of  the  results  of 
abstract  or  conceptual  reasoning  to  the  data  of  experi- 
ence is  in  direct  proportion  to  the  degree  of  indepen- 
dence of  the  relations  dealt  with  from  the  other  relations 
which  constitute  the  conditions  of  real  existence  in  the 
object  represented  in  the  operation  of  thought.    Herein 
lies  the  preeminence  of  geometry  among  the  physical 
sciences.     In  the  physical  sciences  usually  so  called  the 
relations  about  which  these  sciences  are  conversant  are 
closely  interdependent ;  the  thermal,  electric,  magnetic, 
optical,  and  chemical  properties  of  a  body  in  various 
ways  determine  each  other.     If  the  nature  and  degree 
of  this  interdependence   were   accurately  known  and 
could  be  brought  within  the  reach  of  exhaustive  con- 
ceptual analysis,  these  sciences  would  become  deductive 
to  the  same  extent  to  which  the  science  of  geometry  is 
deductive.     All  the  physical  sciences   are   constantly 
striving  to  progress  in  this  direction,  but  the  progress 
is  so  slow  as  to  afford  little  hope  that  the  goal  here  in- 
dicated will  ever  be  reached.     One  reason  for  this  is 


TRANSCENDENTAL  GEOMETRY.  '      227 

that  the  number  of  newly  discovered  relations  multiplies 
at  the  same  (if  not  at  a  greater)  rate  at  which  the  nature 
and  degree  of  the  interdependence  between  the  relations 
already  known  are  brought  to  light.  And  the  difficulty 
of  determining  the  interdependence  in  question  increases 
geometrically  as  the  number  of  new  relations  is  aug- 
mented arithmetically. 

The  foregoing  reflections  are  sufficient,  in  my  judg- 
ment, to  show  that  the  sensationalist  views  of  space  and 
of  the  nature  and  warrant  of  geometrical  truth  are  un- 
tenable, at  least  in  the  form  in  which  they  are  pro- 
pounded by  Mr.  Mill.  But  these  reflections  do  not  in 
the  least  degree  impugn  the  general  proposition  that 
all  our  knowledge  of  the  objective  world  is  derived 
from  experience.  This  proposition  appears  to  me  to  be 
undeniable,  and  is,  doubtless,  assented  to,  explicitly  or 
in  some  mode  of  implication,  by  every  sane  person  at 
the  present  day,  the  only  controversies  respecting  it 
being  disputes  about  the  meaning  of  terms.  But  the 
sensationalists,  and  especially,  as  I  have  already  shown, 
the  founders  and  supporters  of  transcendental  geometry, 
advance  a  thesis  which  is  to  be  carefully  distinguished 
from  the  proposition  just  stated.  They  maintain  that 
space  is  not  only  objectively  real,  but  a  direct  and  inde- 
pendent object  of  sensation  whose  properties  are  to  be 
empirically  ascertained  like  those  of  any  other  physical 
thing.  This  assertion  has  been  met  with  the  counter- 
assertion,  made  by  the  antagonists  of  geometrical  tran- 
scendentalism, that  space,  like  time,  is  not  an  indepen- 
dent object  of  sensation,  but,  as  Kant  has  taught,  or  is 
supposed  to  have  taught,  a  mere  form  of  intuition,  a 
state  or  condition  of  the  intellect  existing  independently 
and  in  advance  of  all  sensible  experience.  The  contest 
between  the  champions  of  the  new  doctrine  and  their 


228  CONCEPTS  OF  MODERN  PHYSICS. 

opponents  has  been  mainly  carried  on  under  the  belief, 
common  to  both  the  contending  parties,  that  these  views 
are  strict  alternatives,  and  that  no  other  view  is  admis- 
sible or  possible.  Let  us  test  these  two  conflicting  as- 
sertions by  facts  of  cognition  about  which  there  is  no 
contest,  or  which  clearly  can  not  be  contested  on  rational 
grounds. 

First,  as  to  the  assertion  of  Riemann  and  Helm- 
holtz  :  if  space  is  a  physically  real  object,  it  certainly  is 
not  a  thing  outside  of,  coordinate  with,  and  different 
from,  other  physical  objects.  When  we  say  that  all 
things  are  in  space,  we  do  not  mean  that  they  are  con- 
tained in  it  as  water  is  contained  in  a  vessel,  but  we 
mean  that  there  is  no  objectively  real  thing  which  is 
not  spatially  extended,  or,  according  to  the  usual  form 
of  expression,  that  spatial  extension  is  a  primary  prop- 
erty of  all  varieties  of  objective  existence.  This  fact 
is  so  obvious  that  Descartes  was  led  by  it  to  maintain 
that  spatial  extension  was  the  only  true  essence  of  ob- 
jective reality.  In  what  way,  then,  and  by  what  means, 
do  we  distinguish  between  space  and  physical  things 
ordinarily  so  called  ?  Certainly  not,  or  at  least  not  di- 
rectly, by  sensation.  Different  acts  of  sensation  may 
present  different  properties  of  the  same  object,  and 
these  properties  may  thus  be  dissociated.  But  no  act 
of  sensation  dissociates  the  extension  of  a  body  from  all 
its  other  properties  and  presents  the  property  of  exten- 
sion alone.  The  sensationalists,  however,  contend  (and 
here  they  trench  upon  the  ground  of  their  opponents, 
the  Kantian  idealists)  that,  although  there  are  no  phys- 
ical objects  without  spatial  extension,  and  although 
such  extension  is  in  a  sense  a  common  property  of  all 
physical  objects,  nevertheless  these  objects  do  not  fill 
all  space,  there  being  pure  space  between  them.  The 


TRANSCENDENTAL  GEOMETRY.  229 

reply  to  this  is  that  this  assertion,  if  true,  does  not  help 
the  sensationalists.  For  acts  of  sensation  are  possible 
only  when  and  where  there  is  objective  difference  and 
change ;  we  have  direct  sensation  of  different  and  vari- 
able physical  qualities  ordinarily  so  called,  and  not  of 
that  which  is  absolutely  homogeneous  and  invariable. 
Here  comes  in  Hobbes's  law :  tientire  semper  idem  et 
non  sentire  ad  idem  recidunt.  It  is  precisely  the  fact 
of  its  homogeneity  and  unchangeableness,  in  addition 
to  that  of  its  invariable  presence  in  all  physical  objects, 
which  distinguishes  the  property  of  spatial  extension 
from  all  other  properties  characteristic  of  a  real  thing, 
and  enables  the  sensationalist  to  speak  of  the  existence 
of  space  at  all.  If  this  distinction  could  be  obliterated 
— if  the  cognitive  or  conceptual  barrier  which  separates 
the  sensations  produced  by  physical  action  from  the 
states  of  consciousness  representative  of  space  were  once 
broken  down — there  would  no  longer  be  any  ground 
whatever  for  the  distinction  between  the  "  properties  " 
of  space  and  the  properties  of  matter  in  any  of  its  va- 
rieties. We  should  be  constrained  to  say  that  the  only 
form  or  variety  of  objective  existence  is  either  space  or 
matter  (it  being  a  mere  question  of  nomenclature 
which),  and  that  all  the  properties  we  now  attribute  to 
matter  are  in  truth  and  in  fact  properties  of  space. 

That  all  this  should  have  escaped  the  attention  of 
Riemann  and  Helmholtz  is  marvelous,  considering  the 
assumption  made  by  both  in  order  to  account  for  the 
alleged  necessity  of  attributing  to  space  a  constant 
measure  of  curvature,  and  thus  limiting  the  number  of 
the  species  of  space,  which,  according  to  their  state- 
ment, are  admissible,  to  three,  viz. :  spherical  space 
with  a  positive  curvature,  pseudo-spherical  space  with  a 
negative  curvature,  and  flat  or  homaloidal  space  with  a 


230  CONCEPTS  OF  MODERN  PHYSICS. 

curvature  equal  to  zero.*  I  allude  to  the  assumption 
that  bodies,  in  the  language  of  Riemann  already  quoted, 
"  exist  independently  of  their  location  in  space,"  which 
means,  of  course,  that  they  are  at  least  different  from, 
if  they  have  not  a  physical  constitution  wholly  inde- 
pendent of,  space.  But  for  this  assumption  there  can 
be  no  valid  reason  founded  upon  or  consistent  with  the 
premisses  of  the  transcendental  theory,  why  space  may 
not  be  essentially  paraboloidal,  or  hyperboloidal,  or 
polyhedral,  or  of  any  other  inherent  form  evolvable 
from  the  creative  fancy  of  the  next  non-horn aloidal  in- 
tellect. 

This  brings  me  to  the  allegation  of  the  transcenden- 
talists,  that  the  properties  of  space,  such  as  the  degree 
and  form  of  its  curvature,  are  to  be  determined  by  ex- 
periment. How  would  such  a  determination  be  ef- 
fected? Suppose  an  astronomer,  at  proper  intervals, 
directed  his  telescope  to  some  fixed  star  whose  distance 
from  the  earth  he  knew  in  some  way  (say  from  spec- 
troscopic  data)  to  be  far  greater  than  that  of  Arcturus, 
for  the  purpose  of  ascertaining  its  parallax.  And  sup- 
pose he  found  this  parallax  sensibly  to  exceed  that  of 
the  less  distant  star — in  other  words,  suppose  he  found 
that  the  angle  of  intersection  between  the  lines  of  his 
vision  was  different  from  that  required  by  the  known 
facts  and  laws  of  astronomy  and  optics  :  what  would  be 
his  conclusion  ?  It  is  not  difficult  to  anticipate  the  an- 
swer to  this  question,  for  the  case  supposed  is  not  with- 
out precedent  in  the  history  of  astronomy.  Displace- 
ments in  the  lines  of  vision  have  repeatedly  been 
observed  by  astronomers,  who  were  unable  to  account 

*  Felix  Klein  "  (Ueber  die  Nicht-Euklidische  Geometric,"  Mathema- 
tische  Annalen,  vol.  iv,  p.  577)  designates  these  kinds  of  space  as  elliptic, 
parabolic,  and  hyperbolic. 


TRANSCENDENTAL  GEOMETRY.  231 

for  them  by  the  facts  and  natural  laws  of  which  they 
had  knowledge.  In  the  early  part  of  the  last  century, 
Bradley  (with  the  aid  of  Molyneux)  made  a  series  of 
telescopic  observations  of  the  star  7  Draconis,  to  the 
end  of  determining  the  amount  of  its  apparent  displace- 
ment due  to  the  orbital  motion  of  the  earth,  so  as  to 
detect  the  annual  parallax  of  the  fixed  stars — an  achieve- 
ment very  desirable  in  Bradley's  time  by  reason  of  a 
standing  objection  to  the  Copernican  system  urged  on 
the  ground  of  the  alleged  absence  of  such  a  parallax. 
To  his  surprise  he  found  a  displacement  different  in 
direction  and  far  greater  in  degree  than  that  looked  for. 
This  anomaly  had  to  be  explained ;  and  Bradley  knew 
of  no  physical  cause  to  which  it  could  be  assigned.  He 
thought  for  a  time  of  nutation  ;  then  of  refraction ;  but 
he  soon  became  satisfied  that  no  explanation  was  af- 
forded by  either  fact.  He  was  finally  led,  by  a  careful 
study  of  the  variations  in  the  direction  and  rate  of  the 
displacement,  to  look  for  a  solution  of  the  mystery  in 
the  composition  between  the  velocity  of  light  and  that 
of  the  earth's  orbital  motion,  and  thus  became  the  dis- 
coverer of  what  is  now  known  as  the  aberration  of  light. 
Amid  all  his  perplexities,  however,  it  does  not  appear 
at  any  time  to  have  occurred  to  him  that  the  anomalous 
phenomenon  could  be  the  result  of  a  constitutional 
crook  in  space.  And  it  may  be  asserted  with  confi- 
dence that  there  is  no  astronomer  living  to-day  who 
would  attribute  the  anomalous  parallax,  whose  dis- 
covery I  have  supposed,  to  a  spatial  pseu do  sphericity. 
For,  irrespective  of  all  other  considerations,  the  astrono- 
mer would  at  once  meet  every  suggestion  of  this  sort 
with  the  objection  that  an  inherent  curvature  of  space 
presupposes  differences  between  its  several  parts — hete- 
rogeneities in  its  internal  constitution  —  and  that  the 


232  CONCEPTS  OF  MODERN  PHYSICS. 

hypothesis  suggested,  therefore,  involved  nothing  less 
than  the  attribution  to  space  of  the  very  properties  by 
the  absence  of  which  alone  it  is  distinguishable  from 
matter. 

The  theory  of  the  geometrical  transcendentalists  is 
thus  invalidated  by  the  absurdity  of  its  fundamental  as- 
sumption. Space  is  not,  can  not  be,  an  object  of  sen- 
sation. The  attribution  to  space  of  relations  and  sen- 
sible interactions  of  the  kind  reflected  in  sensation  is 
impossible  without  the  assumption  of  diversities  among 
its  constituent  parts,  the  denial  of  which  is  the  basis  of 
every  notion  or  concept  of  space,  whatever  may  be  the 
logical  or  psychological  doctrine  to  which  that  notion  is 
referred.  Are  we  driven,  then,  to  the  counter-assump- 
tion of  the  Kantian  idealists,  that  space  is  a  purely  sub- 
jective form  of  intuition  existing  in  the  mind  inde- 
pendently of  and  antecedently  to  all  acts  of  sensa- 
tion— to  the  doctrine  of  the  metaphysical  and  mathe- 
matical adversaries  of  geometrical  transcendentalism  ? 
Let  us  see  by  what  arguments  this  doctrine  is  en- 
forced. 

The  Kantian  idealists  affirm  that  the  idea  of  space  is 
not  only  an  invariable  element  of  every  act  of  sensation, 
but  a  condition  precedent  to  sensation ;  that,  before  we 
are  able  to  refer  any  subjective  impression  to  an  objec- 
tive cause,  and  thus  to  speak  of  the  existence  of  objec- 
tively real  things  or  phenomena  at  all,  the  basis  of  this 
reference — of  the  relation,  not  merely  between  the  With- 
in and  the  Without,  but  also  between  two  elements  at 
least  of  the  Without  whose  interaction  produces  the  sen- 
sation— must  already  be  present  in  the  intellect.  Sensa- 
tion, it  is  said,  is  of  objects ;  it  is  essentially  a  step  from 
a  subjective  affection  or  feeling  to  an  objective  reality. 
Where  is  the  ground  for  this  step  ?  Not,  contends  the 


TRANSCENDENTAL  GEOMETRY.  233 

Kantian,  in  the  world  of  objects ;  for  the  objects  are 
reached  and  become  existent  in  intuition  and  sensation 
only  by  means  of  the  step.  It  must  be  in  the  subject, 
therefore,  in  the  intellect ;  and  it  must  be  present  there 
in  advance  of  the  act  of  sensation.  That  this  is  so,  ap- 
pears, moreover  (it  is  claimed),  from  the  fact  that  the 
idea  of  space  is  absolutely  indestructible.  We  can  men- 
tally evacuate  space  of  its  sensible  contents ;  the  intel- 
lect can  "think  away"  everything  that  is  an  object  of 
sensation  ;  but  it  can  not  "  think  away "  space  itself. 
Space  is  an  integral  part  of  all  states  of  consciousness 
whatever. 

The  foregoing  exposition  is  a  fair  and  sufficiently 
exhaustive  statement  of  the  Kantian  view.  This  view 
has  one  feature  in  common  with  that  of  the  sensation- 
alists to  which  I  have  already  made  incidental  allusion 
— the  assumption  that  space  exists,  either  as  an  object 
of  sensation  or  as  a  form  of  intuition,  as  an  independent 
fact,  and  is  therefore  susceptible  of  objective  or  subjec- 
tive apprehension  ~by  itself.  I  have  already  shown  that 
this  assumption,  in  the  sensationalist  sense,  is  unfounded. 
And,  upon  careful  examination,  it  proves  equally  un- 
founded in  the  sense  of  the  idealists.  It  is  not  true 
that  we  can  mentally  evacuate  space  of  all  its  contents, 
and  have  in  the  mind,  or  before  the  mind,  the  form  or 
image  of  pure  space.  On  the  contrary,  the  idea  of 
space  is  invariably  associated  in  consciousness  with  some 
definite  quality  of  sense.  When  we  attempt  to  bring 
space  before  the  mind  (or,  as  it  is  usually  called,  to 
"realize"  it)  in  its  visual  aspect,  it  always  appears  in 
synthesis  with  a  mental  reproduction  of  some  sensation 
of  color,  however  faint.  Similarly,  when  we  make  the 
effort  of  mentally  "  realizing  "  or  representing  it  in  its 
tactual  aspect,  it  proves  equally  indissociable  from  a 


234  CONCEPTS  OF  MODERN   PHYSICS. 

reproduction  of  some  form  of  pressure  or  feeling.*  In 
this  respect  the  arguments  of  Hume  and  Berkeley 
(which  are  of  necessity  simple  appeals  to  consciousness) 
have  never  been  successfully  met.  The  dissociation  be- 
tween the  "  idea  "  of  spatial  extension  and  the  feeling  or 
feelings  constituting  sensation  which  we  are  able — and, 
for  the  purposes  of  discursive  reasoning,  constrained — 
to  effect,  is  not  an  intuitional,  but  a  conceptual  dissocia- 
tion. Whenever  we  contemplate  and  reason  about  an 
objectively  real  thing,  we  can,  in  virtue  of  the  power  of 
abstraction,  attend  to  the  property  of  spatial  extension, 
in  total  disregard  of  its  sensible  qualities ;  but  whenever 
we  strive  to  bring  its  extension  before  the  mind  as  real 
— to  frame  a  mental  image  of  extension,  or  to  represent 
it  as  a  distinct  form  of  intuition — we  are  instantly  forced 
to  invest  or  associate  it  with  some  one  datum  of  sensa- 
tion which  we  interpret  as  the  incident  or  reaction  of 
a  physical  process.  Intuition  (using  the  word  in  the 
Kantian  sense)  is  an  integral  part  of  sensation,  and  ap- 
pears as  such  alike  in  the  presentations  of  sense  and 
their  representations  or  reproductions  in  the  phantasy. 

This  disposes  of  the  Kantian  argument  that  space 
must  be  a  subjective  form  of  intuition  because  the  mind 
can  not  banish  it  from  consciousness.  And  another 
simple  reflection  is  equally  fatal  to  the  claim  that  space 
must  be  a  subjective  form  existing  in  advance  of  all 
acts  of  sensation,  inasmuch  as  it  is  the  indispensable 
ground  for  the  step  by  which  the  intellect  reaches  an 
object  external  to  itself.  The  obvious  answer  to  this  is 
that,  if  space  is  purely  subjective,  being  wholly  in  the 
mind,  it  certainly  can  not  afford  ground  for  a  step  out 

*  Cf.  Sir  William  Hamilton's  Lectures  on  Metaphysics,  Lect.  xxii ; 
Stumpf,  Ueber  den  psychologischen  Ursprung  der  Raumvorstellungen 
(Leipzig,  Hirzel,  1873),  p.  19. 


TRANSCENDENTAL  GEOMETRY.      '   235 

of  it.  This  reflection  is  the  true  basis  of  the  post- 
Kantian  species  of  idealism,  such  as  that  of  Fichte,  and, 
in  a  sense,  of  Schopenhauer.  But  the  whole  argument, 
as  well  as  the  idealistic  perplexities  that  have  been  oc- 
casioned by  it,  is  founded  on  the  old  ontological  assump- 
tion that  things  or  entities  exist  independently  of  each 
other  and  otherwise  than  as  terms  of  relations.  That 
this  is  not  true  of  objectively  real  things  has  been  suffi- 
ciently enforced  in  the  preceding  pages  of  this  book ; 
and  it  is  equally  untrue  of  the  relation  between  the 
cognizing  subject  and  its  object.  In  every  act  of  pri- 
mary cognition,  the  objective  phenomenon,  so  called, 
and  its  subjective  counterpart  are  born  into  conscious- 
ness at  the  same  moment,  because  the  reality  of  either 
depends  upon  that  of  the  other.  This  is  the  great  pri- 
mary and  irreducible  fact  of  cognition,  which  is  not  the 
less  a  fact  because  it  has  been  misinterpreted  by  the 
metaphysicians  in  a  variety  of  ways,  and  has  given  rise 
to  a  host  of  absurd  cognitive  theories. 

What,  then,  is  the  real  nature  of  space  and  what  is 
the  true  source  of  our  knowledge  respecting  it  ?  If  the 
preceding  considerations  are  valid  and  conclusive,  this 
question  admits  of  but  one  answer.  Space  is  a  concept, 
a  product  of  abstraction.  All  objects  of  our  sensible  ex- 
perience present  the  feature  of  extension  in  conjunction 
with  a  number  of  different  and  variable  qualities  at- 
tested by  sensation ;  and,  when  we  have  successively 
abstracted  these  various  sensations,  we  finally  arrive  at 
the  abstract  or  concept  of  a  form  of  spatial  extension. 
I  purposely  say  form  of  extension,  and  not  simply  ex- 
tension or  space,  for  the  former,  and  not  the  latter,  is  the 
summum  genus  of  the  line  of  abstraction  here  indicated. 
If  the  word  "  concept "  be  used  in  the  sense  in  which  it 
is  representative  of  a  possible  object  of  intuition,  a  spa- 


236  CONCEPTS  OF  MODERN  PHYSICS. 

tidily  extended  form  is  the  last  result  of  the  process  by 
which  an  object  or  phenomenon  can  be  conceived.  The 
abstract  or  concept  (using  the  word  now  in  its  wider 
sense)  extension  generally r,  or  space,  is  reached  by  an- 
other series  of  abstractions  of  which  I  may  have  some- 
thing to  say  hereafter.  The  failure  to  discriminate 
between  those  concepts,  so  called,  which  involve  no 
reference  to  limits  or  forms  and  the  true  summa  genera 
of  the  classification  of  sensible  objects  is  one  of  the 
sources  of  the  confusion  which  everywhere  besets  the 
theory  of  transcendental  space,  as  we  shall  presently 
see. 

The  doctrines  of  the  idealists  (more  properly  called 
intellectualists)  respecting  the  nature  of  space  are,  there- 
fore, as  untenable  as  those  of  the  sensationalists.  And 
the  opinion  of  the  disciples  of  Kant  and  Schopenhauer, 
that  the  teachings  of  transcendental  geometry  can  be 
refuted  by  an  appeal  to  the  "  Transcendental  ^Esthet- 
ics" of  the  "Critique  of  Pure  Reason"  is  a  mistake. 
The  proposition  that  space  is  a  pure  subjective  form  of 
intuition,  if  true,  could  not  in  the  slightest  degree  shake 
the  position  of  the  geometrical  transcendentalist.  His 
simple  retort  upon  the  Kantian  is,  that,  if  space  is  an 
innate  form  or  condition  of  the  intellect  determining 
the  apprehension  of  external  objects  in  a  certain  order, 
or  according  to  certain  laws,  it  is  again  a  question  of 
fact,  what  is  that  order  and  what  are  those  laws. 
"Whether  space  be  within  the  mind  or  without  it,  the 
question  of  its  flatness,  sphericity  or  pseudo-sphericity 
remains.  "Whether  the  form  of  the  lines  and  surfaces 
possible  in  space  is  the  result  of  its  physical  constitution 
outside  of  the  mind,  or  of  the  internal  constitution  of 
the  mind  itself — in  either  case  the  fact  is  the  same, 
whatever  it  may  prove  to  be.  This  is  in  entire  ac- 


TRANSCENDENTAL  GEOMETRY.  "     237 

cord  with  Kant's  own  distinct  declaration  in  his  "  Notes 
to  Transcendental  ^Esthetics,"  *  when  he  says  that  our 
mode  of  intuition  is  not  necessarily  confined  to  the 
peculiar  constitution  of  our  minds,  but  may  be  shared 
by  other  thinking  beings,  "although  this  is  a  matter 
which  we  are  unable  to  decide."  From  this  declaration 
the  inference  is  unavoidable  that  the  question,  what  the 
precise  form  of  intuition  is  in  a  given  intellect,  is  purely 
a  question  of  fact.  In  this  respect,  then,  Helmholtz  f 
is  unquestionably  in  the  right  as  against  Land,  Krause, 
Becker,  and  the  other  Kantians. 

Having  reached  the  conclusion  that  space  is  neither 
a  physical  object  of  sensation,  nor  an  innate  form  of  the 
mind  independent  of  and  preexisting  to  all  sensation, 
but  a  concept,  we  are  now  able  to  enter  upon  a  series 
of  considerations  akin  to  those  already  presented  against 
the  alleged  experimental  determinability  of  the  curva- 
ture of  space,  by  which  the  true  character  of  the  tran- 
scendental theory  of  space  may  be  so  thoroughly  exhib- 
ited that  there  can  be  no  rational  controversy  respecting 
its  merits.  The  first  of  these  considerations  is  this  :  If 
the  doctrines  of  the  transcendentalists  are  founded  in 
fact,  it  follows  that  there  is  in  space  a  coercive  power 
resulting  from  its  constitution  which  makes  lines  and 
surfaces  other  than  those  conforming  to  its  inherent 
figure  impossible.  If  space  is  not  "flat,"  but  spherical, 
for  instance — I  assume  for  the  moment,  and  for  the 
sake  of  argument,  that  there  is  sense  in  the  assertion  of 
the  "flatness"  of  ordinary  "Euclidean"  space — then 
every  line  in  it  necessarily  follows  a  definite  course  to 
which  it  is  astricted  by  the  internal  law  governing  the 

*  Kritik  der  reinen  Vernunft  (ed.  Rosenkranz),  p.  49. 
f  Cf.  "  The  Origin  and  Meaning  of  Geometrical  Axioms,"  Mind,  vol. 
iii,  p.  212  scq.,  also  Die  Thatsachcn  in  der  Wahrnehmung,  Berlin,  1879. 


238  CONCEPTS  OF  MODERN  PHYSICS. 

arrangement  of  its  parts.  From  this  it  is  a  legitimate 
and  inevitable  consequence  that,  in  a  space  of  definite 
and  inherent  curvature,  lines  even  of  different  degrees 
of  curvature  are  impossible.  The  measure  of  curvature 
of  such  a  space  being  once  determined,  all  its  lines  must 
conform  to  it.  To  this  it  is  no  answer  to  say  that  Lo- 
batschewsky  and  Beltrami  have  shown  the  practicability 
of  constructing  consistent  and  logically  coherent  sys- 
tems of  geometry  on  the  principle  of  the  non-parallel- 
ism of  "  shortest  lines,"  and  that  Professor  Lipschitz 
has  demonstrated  that  the  laws  of  motion  as  dependent 
on  motive  forces  could  also  be  consistently  transferred 
to  spherical  or  pseudo-spherical  space,  so  that  the  com- 
prehensive expression  for  all  the  laws  of  dynamics, 
Hamilton's  principle,  may  be  directly  transferred  to 
spaces  of  which  the  measure  of  curvature  is  other  than 
zero.  For  the  constructions  of  Lobatschewsky  and  Bel- 
trami (which  serve  also  as  the  basis  of  Lipschitz's  inves- 
tigations) are  all  constructions  of  lines  and  surfaces ; 
and  these  constructions  are  founded  upon  postulates 
utterly  inconsistent  with  the  properties  of  non-Euclidean 
space.  One  of  these  postulates  is,  that  in  spherical  as 
well  as  in  pseudo  spherical  space  it  is  possible  to  trace 
lines  of  any  degree  of  curvature,  and  therefore  also  lines 
whereof  the  curvature  is  zero,  that  is  to  say,  straight 
lines  in  the  old  sense.  How,  indeed,  could  the  "  meas- 
ure of  curvature "  be  otherwise  determined  ?  That 
measure  depends  upon  the  radius  of  curvature ;  accord- 
ing to  Gauss,  the  measure  of  curvature  belonging  to 
every  surface  that  admits  of  the  motion  of  the  figures 
lying  upon  it,  without  change  of  any  of  their  lines  and 
angles,  measured  along  it,  is  the  constant  reciprocal  of 
the  greatest  and  least  radii  of  curvature.  These  radii 
are  straight  radii,  in  the  old  sense ;  for,  if  they  are  not 


TRANSCENDENTAL  GEOMETRY.     '    239 

straight,  they  are  of  some  definite  degree  of  curvature, 
which  again  can  be  determined  only  by  reference  to 
another  particular  radius,  and  so  on,  either  ad  infinitum, 
or  until  we  come  at  last  to  the  old  Euclidean  straight 
line. 

The  legitimate  premisses  of  the  theory  of  non-Eu- 
clidean space  lead  to  the  inevitable  conclusion  that  the 
lines  of  such  a  space,  though  curves,  have  neither  tan- 
gents nor  normals,  neither  radii  nor  cords,  and  that  on 
the  grounds  of  non-Euclidean  postulates  alone  they  are 
wholly  indeterminable.  This  is  again  a  curious  exem- 
plification of  the  ontological  error  according  to  which 
things  and  forms  are  determinable  in  themselves,  with- 
out reference  to,  or  contrast  with,  correlative  things 
and  forms.  What  is  especially  remarkable,  in  this 
aspect  of  the  doctrine  of  the  transcendentalists,  is  the 
ascription  to  real  space  of  an  inherent  disjunction  be- 
tween the  forms  of  its  alleged  curvature — the  assertion 
that  its  measure  of  curvature  must  be  either  positive,  or 
negative,  or  zero.  This  assertion  is  all  the  more  re- 
markable by  reason  of  the  transcendentalist  claim  that 
the  new  doctrine  has  emancipated  the  old  system  of 
geometry  from  its  arbitrary  limitations,  and  is  a  widen- 
ing, a  logical  expansion,  of  the  idea  of  space. 

The  source  of  all  the  perplexities  in  which  we  find 
ourselves  involved  by  the  assumptions  and  theories  of 
the  transcendentalists  is  so  obvious,  that  it  is  a  wonder 
how  it  has  come  to  be  completely  ignored  by  the  adver- 
saries of  the  new  doctrine  no  less  than  by  its  adherents. 
The  parent  error  of  this  doctrine  is  the  assertion  that 
the  space,  with  which  ordinary  "  Euclidean  "  geometry 
deals,  is  a  "  flat,"  and  not  a  spherical  or  pseudo-spherical 
space.  The  truth  is  that  the  space  whose  idea  or  notion 
underlies  all  geometrical  constructions  whatever^  in- 


240  CONCEPTS  OF  MODERN  PHYSICS. 

eluding  those  of  the  pangeometers,  is  neither  flat,  nor 
spherical,  nor  pseudo-spherical,  nor  of  any  other  inher- 
ent figure,  but  is  simply  the  intuitional  and  conceptual 
possibility  of  tracing  any  or  all  of  the  lines  characteris- 
tic of  plane,  spherical,  ellipsoidal,  pardboloidal,  hyper- 
~boloidal,  etc.,  and,  to  some  extent,  pseudo-spherical  sur- 
faces within  it — a  possibility  due  to  the  circumstance 
that  it  is  nothing  more  nor  less  than  a  concept  formed 
by  dismissing  from  our  mental  representation  of  physi- 
cal objects,  not  only  all  the  attributes  constituting  their 
physical  properties  other  than  extension,  but  also  all 
the  determinations  of  figure  by  which  they  are  distin- 
guished. This  is  the  only  sense  in  which  we  have  any 
right  to  speak  of  space  as  even  or  homaloidal.  Space 
has  no  internal  structure  or  inherent  figure,  because  it 
is  not  a  physical  object,  and  therefore  has  no  "  proper- 
ties "  which  can  be  ascertained  by  experiment  or  obser- 
vation. Nor  has  it  any  properties,  rightly  so  called, 
that  are  determinable  a  priori,  by  an  act  of  intuition. 
Space  is  one  of  those  ultimates  of  abstraction  in  which 
the  connotation  coincides  with  the  denotation,  and  in 
which,  therefore,  true  connotation  is  at  an  end.  I  re- 
peat: space  has  no  properties,  for,  considered  as  an 
entity,  it  has  no  relations,  its  very  essence  being  a  denial 
of,  or  abstraction  from,  all  relations.  For  this  reason  it 
is  an  abuse  of  terms  to  define  geometry  (as  is  so  fre- 
quently done,  and  has  lately  been  done  by  Professor 
Henrici  *)  as  "  the  science  whose  object  it  is  to  investi- 
gate the  properties  of  space."  The  object  of  geometry  is 
the  investigation  of  the  possible  determinations  or  limi- 
tations of  space,  i.  e.,  of  the  relations  between  the  vari- 
ous forms  of  extension  or  of  the  properties  of  figures.f 

*  Encycl.  Britan.,  s.  v.  Geometry. 

f  In  this  sense  D'Alembert  (Elemens  de  Philosophic,  §  16— (Euvres, 


TRANSCENDENTAL  GEOMETRY.  24:1 

The  whole  science  of  geometry  is  conversant  about  that 
which  the  concept  space  of  necessity  excludes,  viz.,  about 
determinations  or  limits.  Geometry,  indeed,  has  refer- 
ence to  space,  inasmuch  as  the  determinations  with 
which  it  deals  are  spatial  determinations.  Upon  this  fact 
arises  the  difference  between  the  scope  of  geometry  and 
that  of  the  other  branches  of  pure  mathematics,  and  the 
inapplicability  of  many  of  the  methods  and  results  of 
mathematical  analysis  to  the  relations  between  the  forms 
of  space — a  difference  the  disregard  of  which  is  prolific 
of  so  many  errors  in  the  reasoning  of  those  who  seek  to 
draw  conclusions  respecting  the  "  properties  "  of  space 
(such  as  the  possible  number  of  its  dimensions)  from 
the  abstract  concept  quantity.  Geometry  is  undoubt- 
edly an  empirical  science,  though  not  in  the  sense  in 
which  the  term  "empirical"  is  generally  understood, 
and  especially  not  in  the  sense  in  which  it  is  interpreted 
by  Mill  and  the  geometrical  transcendentalists.  It  is  an 
empirical  science,  inasmuch  as  it  deals  with  a  property 
of  physical  things,  extension,  which  is  an  ultimate,  or, 
rather,  primary  and  irreducible  datum  of  the  act  of  sen- 
sation—just as  much  such  a  datum  as  the  sensation  of 
color  with  which,  as  I  have  shown,  the  visual  intuition 
of  space  is  invariably  associated.  All  attempts,  such  as 
those  of  Herbart,  to  produce  the  "  idea  "  of  extension 
by  an  elaboration  of  such  data  of  sensation  as  are  com- 
monly designated  as  qualitative,  are  as  abortive  as  the 
corresponding  attempts  to  deduce  the  qualitative  ele- 
ments of  sensation  from  forms  of  extension.  The  pri- 
mary datum  of  extension  is  the  empirical  element  in  the 
science  of  geometry.  This  primary  datum  is  not  space, 
but  limited  extension,  for  sensation  and  intuition  are  of 

tome  i,  p.  268)  defines  geometry  as  "  la  science  des  propriety's  de  Peten- 
due  en  lant  qu'on  la  considere  comme  simplement  etendue  etfiguree." 


242  CONCEPTS  OF  MODERN  PHYSICS. 

particular  bodies,  and  therefore  of  limited  extension, 
not  of  extension  generally,  or  space.  Forms  of  limited 
extension,  however,  give  rise  to  the  concept  space,  by 
the  application  of  the  processes  of  abstraction  I  have 
indicated.  On  the  other  hand,  the  conclusions  of  ge- 
ometry are  not  derived  from  empirical  data  alone,  and 
are  not  reached  by  processes  of  induction,  as  Mill  con- 
tends, and  in  that  sense  geometry  is  not  an  empirical 
science.  Nor  is  there  any  geometrical  axiom  which 
is  purely  a  datum  of  sensation,  as  is  asserted  "by  the 
sensationalists,  or  of  intuition,  according  to  the  teach- 
ings of  the  idealists  or  intellectualists.  All  the  geomet- 
rical axioms,  which  serve  as  starting-points  of  deduction, 
contain  two  elements :  an  element  of  intuition  (as  a 
part  of  sensation)  and  an  element  of  arbitrary  intellect- 
ual determination  which  is  called  definition.  The  facts 
of  extension  and  its  limits — surfaces,  lines,  and  points — 
are  given  in  intuition ;  without  sensible  experience  we 
should  not  know  anything  about  geometrical  solids, 
surfaces,  lines,  and  points;  but  nothing  is  deducible 
from  the  existence  of  these  elements,  or  our  intuition  of 
them,  until  they  are  denned.  This  is  evident  upon  a 
simple  inspection  of  the  geometrical  axioms.  The  axiom 
that  between  two  points  but  one  straight  line  can  be 
drawn  (or,  what  is  the  same  thing,  that  two  straight 
lines  can  not  inclose  a  space)  involves  the  definition  of 
the  straight  line — a  definition,  by  the  way,  far  more 
difficult  on  purely  geometrical  grounds  than  that  of  par- 
allels.* Again  :  the  axiom  respecting  parallels,  in  the 

*  The  real  source  of  this  difficulty  lies  in  a  fundamental  defect  of  the 
current  theories  of  cognition — in  the  failure  to  see  that  all  processes  of 
deductive  reasoning  involve  an  ultimate  reference  to  primary  constants 
which  are  not  given  in  experience,  but  established  by  the  intellect.  This 
primary  constant  in  geometry  is  the  straight  line,  or  simple  direction. 
That  the  difficulties  presented  by  the  10th  Euclidean  axiom  ("  two  straight 


TRANSCENDENTAL  GEOMETRY.  243 

form  now  generally  given  to  it,  viz.,  that  through  a 
given  point  but  one  straight  line  can  be  drawn  parallel 
to  another  straight  line,  presupposes  the  definition,  not 
only  of  the  straight  line,  but  of  parallelism  which,  in 
elementary  geometry,  presents  the  difficulty  of  involv- 
ing the  concept  of  infinite  extension,  and  has  given  rise 
to  innumerable  quandaries  (such  as  that  of  infinitely 
distant,  and  yet  real,  points  of  intersection),  among 
which  those  of  the  pangeometrical  sort  are  not  the 
least.  The  Euclidean  list  of  definitions,  postulates  and 
axioms  is  vitiated,  not,  or,  at  least,  not  only,  by  the  fact 
that  his  lines  of  distinction  between  these  several  pre- 
requisites of  geometrical  reasoning  are  not  correctly 
drawn — that  he,  confounds  definitions  with  axioms  and 
postulates  with  both,*  and,  besides,  fails  to  discriminate 
between  axioms  of  quantity  in  general  and  axioms  of 
spatial  quantity — but  by  his  ignorance  or  disregard  of 
the  fact,  to  which  I  have  referred,  that  every  axiom, 
which  is  geometrically  fertile,  involves  a  definition. 

lines  can  not  inclose  a  space  ")  are  of  the  same  nature  with  those  of  the 
12th  (usually  called  the  llth — the  axiom  of  parallelism)  has  long  been 
known.  "  La  definition  et  les  proprieties  de  la  ligne  droite,"  says  D'Alem- 
bert  (Elemens  de  Philosophic,  §  12— (Euvres,  tome  i,  p.  280)  "  ainsi  que 
des  lignes  paralleles  sont  done  Pecueil  et,  pour  ainsi  dire,  le  scandale 
des  elemens  de  geometric." 

*  Hankel  (Vorlesungen  ueber  die  complexen  Zahlen  und  ihre  Func- 
tionen,  p.  52)  draws  attention  to  the  fact  that  the  confusion  above 
referred  to  is  chargeable,  not  to  Euclid,  but  to  his  editors  and  commen- 
tators. "In  all  the  manuscripts,"  says  Hankel,  "which  F.  Peyrard  has 
collected  in  preparing  his  excellent  edition  of  Euclid  ((Euvres  d'Euclide 
trad,  en  Latin  et  en  Fran9ais,  tome  i,  p.  454)  the  famous  llth  principle 
of  the  theory  of  parallels  appears,  not  among  the  Kotvai  evvoiai  relating 
to  equal  and  unequal  quantities,  but  as  the  5th  postulate  (a?Tr)/j.a).  Simi- 
larly the  10th  axiom  in  all  these  manuscripts  appears  as  the  4th  postulate, 
while  the  MSS.  vary  in  respect  to  the  12th  axiom,  it  being  thus  evident 
that  the  three  axioms  owe  the  place  which  unaccountably  they  still  oc- 
cupy in  the  list  of  axioms  to  a  misunderstanding." 


214  CONCEPTS  OF  MODERN  PHYSICS. 

And  this  ignorance — very  excusable  in  Euclid's  time — 
unfortunately  appears  to  be  snared  by  the  writers  of 
geometrical  text-books  at  the  present  day. 

One  of  the  points  upon  which  the  debate  between 
Helmholtz  and  his  opponents  has  largely  turned  is 
the  question  whether  or  not  Beltrami's  pseudo-spherical 
space  is  conceivable  or  imaginable  (vorstellbar) ;  and,  to 
maintain  the  affirmative,  Helmholtz  propounds  a  re- 
markable definition  of  imaginability.  He  defines  the 
power  of  imagining  spatial  forms  as  "the  power  of 
fully  representing  the  sense-impressions  which  the  ob- 
ject would  excite  in  us  according  to  the  known  laws  of 
our  sense-organs  under  all  conceivable  conditions  of  ob- 
servation, and  by  which  it  would  be  distinguished  from 
other  similar  objects."  *  Whatever  may  be  the  general 
merits  of  this  definition,  it  is  certainly  obnoxious  to  the 
charge  of  irrelevancy.  As  the  old  logicians  would  say, 
it  is  founded  upon  an  ignoratio  elenchi,  a  misapprehen- 
sion of  the  question.  Granting,  for  the  sake  of  argu- 
ment, that  the  act  of  imagining  a  form  of  space  is  truly 
described  as  an  anticipation  of  sense-impressions,  the 
question  as  to  the  existence  of  the  power  sought  to  be 
defined  is,  not  what  would  be  the  nature  of  these  im- 
pressions, but  whether  or  not  they  could  coexist  in  the 
imagination  in  the  required  spatial  order  and  form  ac- 
cording to  the  known  laws  of  the  representative  faculty, 
Helmholtz  refers  to  the  attempt  of  Beltrami  to  make 
pseudo-spherical  space  representable  by  projecting  its 
points,  lines  and  surfaces  upon  the  interior  of  an  ordi- 
nary spherical  surface  "  whose  points  correspond  to  the 
infinitely  distant  points  of  pseudo- spherical  space,"  and 
claims  that  this  attempt  is  successful.  In  the  same 
sense  Professor  Sylvester,  in  the  note  to  his  Exeter 

*  "  Origin  and  Meaning  of  Geometrical  Axioms,"  Mind,  vol.  iii,  p.  216. 


TRANSCENDENTAL  GEOMETRY.         245 

address  already  quoted,  observes  that  "every  perspec- 
tive representation  of  figured  space  of  four  dimensions 
is  a  figure  in  real  space,  and  that  the  properties  of  fig- 
ures admit  of  being  studied  to  a  great  extent,  if  not 
completely,  in  their  perspective  representations."  And 
it  has  become  a  standing  assertion  of  the  pangeometers 
that  the  forms  of  a  space  of  any  given  dimension  may 
be  projected  into  the  space  of  the  next  lower  dimension. 
But  this  assertion,  at  best,  holds  good  only  for  the  limits 
of  projection,  where  the  resulting  point  or  figure  ceases 
to  be  an  explicit  reproduction  of  the  figure  projected. 
When  a  straight  line  is  projected  orthogonally  upon 
another  straight  line  at  right  angles  to  it,  it  appears  as 
a  point;  a  form  of  the  first  dimension  is,  in  a  sense, 
reduced  to  the  dimension  zero.  But  the  representative 
point,  by  itself,  does  not  enable  us  to  reproduce,  and 
reason  about  the  line  whereof  it  is  the  projection.  It 
may  be  said  that  we  can  at  least  know  that  the  line  pro- 
jected is  straight ;  but  that  is  a  conclusion  which  fol- 
lows only  from  the  properties  of  lines  as  they  are  other- 
wise known ;  from  the  mere  inspection  of  the  point  it 
is  not  even  inferable  that  it  is  a  projection  of  a  line  at 
all.  Similarly  a  plane  may  be  projected  upon  another 
plane,  so  as  to  appear  as  a  line,  a  form  of  two  dimen- 
sions being  reduced  to  a  form  of  one  dimension  ;  but  it 
does  not  follow  that  we  may  study  the  properties  of  the 
plane  by  merely  contemplating  or  analyzing  the  line. 
The  so-called  projections  of  solids  upon  surfaces  are  in 
fact  projections,  upon  a  normal  surface,  of  several  sur- 
faces making  different  angles  with  it,  and  the  inferences 
from  such  a  projection  respecting  the  properties  of  geo- 
metrical solids  depend  upon  our  associations  of  visual 
with  tactual  impressions  in  which  our  apprehension  of 
geometrical  solidity  has  its  origin.  There  being  con- 


246  CONCEPTS  OF   MODERN  PHYSICS. 

fessedly  no  tactual  or  other  impressions  evidencing  the 
existence  of  a  fourth  dimension,  the  analogy  upon 
which  the  alleged  imaginability  of  transcendental  space- 
forms  is  founded  is  without  support. 

But  it  is  of  little  consequence  what  ground  there  is 
for  the  claim  (which  has  recently  been  urged  in  another 
form  by  Felix  Klein  *)  that  the  resources  of  projective 
geometry  are  sufficient  to  enable  us  to  represent  the 
properties  of  a  space  of  more  than  three  dimensions  in 
tridimensional  space  :  for  the  question  of  representabil- 
ity  is  wholly  foreign  to  the  matter  in  dispute.  If  it 
were  shown,  for  instance,  that  a  pseudo-spherical  sur- 
face may  be,  mentally  or  really,  traced  in  space,  this 
certainly  would  not  prove,  or  tend  to  prove,  that  space 
is  inherently  pseudo-spherical.  There  is  no  doubt  about 
the  imaginability  of  a  spherical  surface,  but  from  this 
it  does  not  follow  that  space  itself  is  spherical.  To 
support  the  conclusion  of  the  immanent  pseudo-sphe- 
ricity of  space  it  would  be  necessary  to  maintain  that 
none  but  pseudo-spherical  surfaces  can  exist,  and,  there- 
fore (conformably  to  the  teachings  of  sensationalism), 
be  represented,  or  imagined  as  existing,  in  it.  And,  in 
view  of  this,  the  whole  argument  of  Helmholtz  not  only 
ceases  to  be  available  as  a  support  of  geometrical  tran- 
scendentalism, but  recoils  upon  himself.  If  pseudo- 
spherical  surfaces  can  be  imagined  to  exist,  and  there- 
fore, upon  his  own  principles,  are  possible  in  "flat" 
space,  why  can  not  ordinary  straight  lines  and  flat  sur- 
faces exist  in  pseudo-spherical  space  ?  And  what,  then, 

*  "  Ueber  die  Nicht-Euklidische  Geometric."  Math.  Ann.,  vol.  iv,  p. 
573.  In  this  article,  as  in  nearly  all  the  writings  of  the  pangeometers, 
who  deal  with  imaginary  and  infinitely  distant  points  ad  libitum,  analyti- 
cal representability  (by  means  of  symbols  among  which  infinite  and 
imaginary  elements  are  treated  as  coordinate  with  real  elements)  is  con- 
founded with  imaginability. 


TRANSCENDENTAL  GEOMETRY.  4.7 

becomes  of  his  telescopic  test  of  the  curvature  of  space  ? 
Or  am  I  under  a  misapprehension  as  to  Helmholtz's 
true  meaning? — does  he  simply  contend  that  pseudo- 
spherical  surfaces  would  be  imaginable  by  pseudo-spher- 
ical beings  with  pseudo-spherical  organs  of  sense,  and 
consequent  pseudo-spherical  intellects  in  a  pseudo- 
spherical  space,  if  it  existed?  That  is  a  proposition 
which  even  Land  and  Krause  would  hardly  dispute. 

The  history  of  cognition  affords  no  illustration,  per- 
haps, of  the  irrepressibility  of  intellectual  traditions 
which  is  more  instructive  than  the  doctrines  of  tran- 
scendental geometry.  Glancing  back  at  the  contents 
of  the  present  chapter,  we  see  that  even  the  science  of 
mathematics — the  exactest  of  all  the  sciences,  whose 
methods  are  said  to  be  as  infallible  as  its  foundations 
are  supposed  to  be  permanent,  and  which,  ever  since 
the  dawn  of  human  intelligence,  has  pursued  the  even 
tenor  of  its  way  amid  all  the  vicissitudes  of  speculation 
— is  not  exempt  from  the  prepossessions  of  ontological 
realism.  The  same  hypostasis  or  reification  of  concepts, 
which  has  given  rise  to  the  atomo-mechanical  theory  in 
physics,  has  led  to  the  doctrine  of  pangeometry  in  math- 
ematics. The  hypostasis  of  space,  by  the  mathemati- 
cians, is  a  strict  analogue  of  the  hypostasis  of  mass  and 
motion  by  the  physicists. 

The  full '  extent,  however,  to  which  the  minds  of 
contemporary  mathematicians  are  bewildered  by  the 
false  light  of  ontology  can  be  brought  into  still  clearer 
view  by  a  further  examination  of  the  speculative  back- 
ground of  transcendental  geometry,  as  it  appears  in  the 
famous  essay  of  Riemann  already  referred  to. 


CHAPTEK  XIY. 

METAGEOMETEICAL    SPACE    IN    THE    LIGHT   OF    MODERN 
ANALYSIS. — KIEMANN'S  ESSAY. 

THE  essay  of  Bernhard  Riemann,  "  On  the  Hypothe- 
ses which  lie  at  the  Base  of  Geometry,"  owes  its  great 
celebrity  to  the  fact  that  he  was  a  mathematical  analyst 
of  the  first  order,  one  of  the  favorite  pupils  of  Gauss, 
under  the  inspiration  of  whose  teachings,  if  not  at  his 
suggestion,  the  essay  was  written — by  whom,  in  fact,  it 
was  presented,  in  1854,  shortly  before  his  (Gauss's) 
death  to  the  philosophical  faculty  of  Goettingen,  and 
by  whom  its  cardinal  propositions  were  expressly  in- 
dorsed as  an  exposition  of  his  own  speculative  opinions. 
Every  intelligent  reader  of  this  essay  will  agree  with 
me,  I  think,  that  its  intrinsic  merit  is  not  at  all  com- 
mensurate with  the  attention  with  which  it  was  re- 
ceived and  the  interest  with  which  it  is  still  generally 
considered.  Not  only  are  its  statements,  both  of  the 
problem  and  of  the  proposed  methods  of  solution,  crude 
and  confused,  but  they  bear  the  impress  throughout  of 
Riemann's  very  imperfect  acquaintance  with  the  nature 
of  logical  processes  and  even  with  the  import  of  logical 
terms.  It  is  apparent,  from  the  whole  tenor  of  the 
essay,  that  its  author  was  an  utter  stranger  to  the  dis- 
cussions respecting  the  nature  of  space  which  have  been 
BO  vigorously  carried  on  by  the  best  thinkers  of  our 
time  ever  since  the  days  of  Kant,  and  that  he  was  so 


RIEMANN'S  DISSERTATION.  249 

little  familiar  with  the  history  of  logic  as  to  be  with- 
out the  faintest  suspicion  of  the  manifold  ambiguity 
of  such  terms  as  "concept"  and  "quantity,"  and  of 
the  necessity  of  their  exact  definition  preliminary  to 
an  inquiry  respecting  the  very  foundations  of  human 
knowledge.* 

The  general  argument  of  the  essay  is,  that  the  nat- 
ure of  space  is  to  be  deduced  from  its  concept ;  that 
the  formation  of  such  a  concept  of  necessity  involves 
its  subsumption  under  a  higher  concept ;  that  this  higher 
concept  is  that  of  a  "multiply  extended  quantity;" 
that,  in  order  to  determine  how  many  kinds  of  space 
are  possible,  it  is  requisite  to  ascertain  in  how  many 
ways  quantity  may  be  "  multiply  extended  "  (mehrfach 
ausgedehnt) ;  and  that,  after  the  number  of  conceptually 

*  Riemann  himself  modestly  apologizes  for  the  philosophical  short- 
comings of  bis  essay  on  the  ground  of  his  inexperience  in  philosophical 
matters.  But  the  crudeness  of  his  speculations  affords  a  very  striking 
illustration,  in  my  judgment,  of  the  well-known  fact  that  exclusive  devo- 
tion to  the  labors  of  the  mathematical  analyst  has  a  tendency  to  develop 
certain  special  powers  of  the  intellect  at  the  expense  of  its  general  grasp 
and  strength.  Although  Sir  William  Hamilton,  no  doubt,  overstated  the 
case  against  the  mathematicians,  I  believe  that  his  suggestions  are  not 
wholly  unworthy  of  attention,  and  that  there  is  force  in  the  words  of 
D'Alembert  (referred  to  by  Sir  William  Hamilton),  which  it  is  perhaps 
safest  to  quote  in  the  original,  without  translation :  "  II  semble  que  les 
grands  geometres  devraient  etre  excellens  metaphysiciens,  au  moins  sur 
les  objets  dont  ils  s'occupent ;  cependant  il  s'en  faut  bien  qu'ils  le  soient 
toujours.  La  logique  de  quelques  uns  Centre  eux  est  renfermee  dans  leurs 
formules  et  ne  s'etend  pas  au  dela.  On  peut  les  comparer  a  un  homme 
qui  aurait  le  sens  de  la  vue  contraire  a  celui  du  toucher,  ou  dans  lequel 
le  second  de  ces  sens  ne  se  perf ectionnerait  qu'aux  depens  de  1'autre.  Cos 
mauvais  metaphysiciens  dans  une  science  ou  il  est  si  facile  de  ne  le  pas 
etre,  le  seront  a  plus  forte  raison  infailliblement,  comme  I'exp6rience  le 
prouve,  sur  les  matieres  ou  ils  n'auront  pas  le  calcul  pour  guide.  Ainsi 
la  geometric  qui  mesure  les  corps,  peut  servir  en  certains  cas  a  mesurer 
les  esprits  m6me."  D'Alembert,  Elemens  de  Philosophic,  §  11 ;  CEuvres, 
tome  i,  p.  276. 


250  CONCEPTS  OF  MODERN  PHYSICS. 

possible  varieties  or  species  of  multiple  extension  lias 
thus  been  fixed,  it  is  a  matter  of  experimental  determi- 
nation which  of  these  varieties  or  species  is  represented 
by  our  space,  i.  e.,  by  the  space  in  which  the  world,  as 
we  know  it,  has  its  being.  After  thus  asserting  that 
the  concept  "  space "  is  to  be  subsumed  under  the  con- 
cept "  quantity,"  Riemann  proceeds  to  declare  that  all 
quantities  are  in  their  nature  multiples  or  aggregates 
(Mannigfaltigkeiteri)  which  are  continuous  whenever 
there  is  continuous  transition  from  one  of  their  several 
"  specializations  "  to  the  other,  and  discrete,  when  there 
is  no  such  transition  ;  that  the  "  specializations"  of  dis- 
crete quantities  are  called  points,  and  those  of  continu- 
ous quantities  elements  /  and  that  continuous  quantities 
are  determined  by  measurement,  while  discrete  quan- 
tities are  determined  by  numeration.  Space,  according 
to  Riemann,  though  a  continuous  quantity,  is  a  quan- 
tity of  Ti-fold  (geometrical)  extension,  and  is  thus  a 
Multiple  or  Aggregate,  and  therefore  a  quantity,  not- 
withstanding its  continuity.  The  degree  of  the  multi- 
plicity of  this  extension — i.  e.,  the  fact  of  its  being  sim- 
ple, twofold,  threefold,  or  generally  n-fold — determines 
the  (logical)  extension  of  the  concept  space. 

We  have  here  five  distinct  propositions,  which,  for 
convenience  of  reference  and  discussion,  may  be  stated 
in  distinct  form  as  follows : 

1.  That  the  nature  of  space  is  to  be  deduced  from 
its  concept. 

2.  That  the  concept  of  space  can  be  formed  and 
determined  only  by  its  subsumption  under  a  higher 
concept. 

3.  That  our  space  is  a  "triply  extended  Multiple  or 
Aggregate,"  the  higher  concept  under  which  its  con- 
cept is  to  be  subsumed  being  that  of  an  "  n-i old  ex- 


RIEMANN'S  DISSERTATION.  251 

tended  Multiple  "  or  a  "  multiply  extended  Aggregate  " 
(eine  n-fach  ausgedehnte  Mannigfaltig~keit\  and  that — 
translating  Riemann's  phraseology  into  its  plain  logical 
import — the  (logical)  extension  of  this  higher  concept 
determines  the  number  of  the  possible  kinds  of  space. 

4.  That  the  conceptual  possibility  of  space  is  coex- 
tensive with  its  empirical  possibility,  though  not  with  its 
empirical  reality. 

5.  That  continuous  quantities  are  coordinate  with 
discrete  quantities,  i.  e.,  are  species  of  the  same  genus, 
both  being  in  their  nature  multiples  or  aggregates.* 

*  The  order  and  numeration  of  these  propositions  is,  of  course,  my 
own ;  in  Riemann's  essay  they  appear  in  very  promiscuous  order.  In 
proof  of  the  general  correctness  of  my  statement  of  Riemann's  doctrines, 
it  is  perhaps  well  to  quote  the  introductory  part  of  his  essay  in  the  origi- 
nal, italicizing  the  more  important  passages  : 

"  Ueber  die  Hypothesen  welche  der  Geometric  zu  Grunde  liegen. 
"  Plan  der  Untersuchung. 

"  Bekanntlich  setzt  die  Geometrie  sowohl  den  Begriff  des  Raumes, 
als  die  ersten  Grundbegriffe  fuer  die  Constructionen  ini  Raume  als  etwas 
Gegebenes  voraus.  Sie  giebt  von  ihnen  nur  Nominaldefinitionen,  waeh- 
rend  die  wesentlichen  Bestiramungen  in  Form  von  Axiomen  auftreten. 
Das  Verhaeltniss  dieser  Voraussetzungen  bleibt  dabei  im  Dunkeln ;  man 
sieht  weder  ob  und  in  wie  weit  ihre  Verbindung  nothwendig,  noch  a  pri- 
ori, ob  sie  moeglich  ist. 

"  Diese  Dunkelheit  wurde  auch  von  Euklid  bis  Legendre,  um  den  be- 
ruehmtesten  ncuereu  Bearbeiter  der  Geometrie  zu  nennen,  weder  von  den 
Mathematikern,  noch  von  den  Philosophen,  welche  sich  damit  beschaeftig- 
ten,  gehoben.  Es  hatte  dies  seinen  Grund  wohl  darin,  dass  der  allge- 
meine  Begriff  mehrfach  ausgedehnter  Groessen,  unter  welchen  die  Raum- 
groessen  enthalten  sind,  ganz  unbearbeitet  blicb.  Ich  habe  mir  daher 
zunaechst  die  Aufgabe  gestellt,  den  Begriff  einer  mehrfach  ausgedehnten 
Groesse  aus  allegemeinen  Groessenbcgriffen  zu  construiren.  Es  wird  daraus 
hervorgehen  dass  eine  mehrfach  ausgedehnte  Groesse  vei'schiedener  Maass- 
verhaeltnisse  faehig  ist,  und  der  Raum  also  nur  cinen  besondern  Fall  einer 
dreifach  ausgedehnten  Groesse  bildet.  Hiervon  aber  ist  eine  nothwendige 
Folge,  dass  die  Saetze  der  Geometrie  sich  nicht  aus  allgemeinen  Groessen- 
begriffen  ableiten  lassen,  sondern  dass  diejenigen  Eigenschaften,  durch 


252  CONCEPTS  OF  MODERN  PHYSICS. 

I  proceed  to  consider  these  propositions  in  their 
order. 

1.  The  first  proposition  is  in  plain  words  a  statement 
of  the  general  ontological  fallacy  (discussed  at  length  in 

welche  sich  der  Raum  von  andern  denkbaren  dreifach  ausgedehnten 
Groessen  unterscheidet,  nur  aus  der  Erfahrung  entnommen  werden  koen- 
nen.  Hieraus  ensteht  die  Aufgabe,  die  einfachsten  Thatsachen  aufzu- 
suchen,  aus  denen  sich  die  Haassverhaeltnisse  des  Raumes  bestimmen 
lassen — eine  Aufgabe,  diederNatur  der  Sache  nach  nicht  voellig  bestimmt 
ist ;  denn  es  lassen  sich  mehrere  Systeme  einfacher  Thatsachen  angeben, 
welche  zur  Bestimmung  der  Maassverhaeltnisse  des  Raumes  hinreichen ; 
am  wichtigsten  ist  fuer  den  gegenwaertigen  Zweck  das  von  Euklid  zu 
Grunde  gelegte.  Diese  Thatsachen  sind,  wie  alle  TJiatsachen,  nicht  noth- 
wendig,  sondern  nur  von  empirischer,  Gewissheit,  sie  sind  Hypothesen,  man 
kann  also  ihre  Wahrscheinlichkeit,  welche  innerhalb  der  Grenzen  der  Beo- 
bachtung  allerdings  schr  gross  ist,  untersuchen,  und  hienach  ueber  die 
Zulaessigkeit  ihrer  Ausdehnung  jenseits  der  Grenzen  der  Beobachtung 
sowohl  nach  der  Seite  des  Unmessbargrossen,  als  nach  der  Seite  des  Un- 
messbarkleinen  urtheilen. 

"  I.  Begriff  einer  n-fach  ausgedehnten  Groesse. 

"  Indem  ich  nun  von  diesen  Auf  gaben  zunaechst  die  erste,  die  Entwicke- 
lung  des  Begriffes  mehrfach  ausgedehnter  Groessen,  zu  loesen  versuche, 
glaube  ichum  somehr  auf  eine  nachsichtige  Beurtheilung  Anspruch  mach- 
en  zu  duerf  en,  da  ich  in  dergleichen  Arbeiten  philosophischer  Natur,  wo  die 
Schwierigkeiten  mehr  in  den  Begriffen,  als  in  den  Constructionen  liegen, 
wenig  geuebt  bin  und  ich  ausser  einigen  ganz  kurzen  Andeutungen  welche 
Herr  Hofrath  Gauss  in  der  zweiten  Abhandlung  ueber  die  biquadratischen 
Reste,  in  den  Goettingischen  gelehrten  Anzeigen,  und  in  seiner  Jubi- 
laeumschrift  darueber  veroeffentlicht  hat,  und  einigen  philosophischen 
Untersuchungen  Herbart's  durchaus  keine  Vorarbeiten  benutzen  konnte. 

"  Groessenbegriffe  sind  nur  da  moeglich,  wo  sich  ein  allgemeiner  Begriff 
vorfindet,  der  verschiedene  Bestimmungsweisen  zulaessL  Je  nacJujem  unter 
diesen  Bestimmungsweisen  von  einer  zu  einer  andern  ein  stetiger  Ueber  gang 
stattfindet  oder  nicht,  bilden  sie  eine  stetige  oder  discrete  Mannigfaltigkeit  ; 
die  einzelnen  Bestimmungsweisen  heissen  im  ersten  Fall  Puncte,  in  letzterem 
Elemente  dieser  Mannigfaltigkeit.  Begriffe,  deren  Bestimmungsweisen 
eine  discrete  Mannigfaltigkeit  bilden,  sind  so  haeufig,  dass  sich  fuer  be- 
liebig  gegebene  Dinge  wenigstens  in  den  gebildeteren  Sprachen  immer 
ein  Begriff  auffinden  laesst,  unter  welchem  sie  enthalten  sind  (und  die 
Mathematiker  konnten  daher  in  der  Lehre  von  den  discreten  Groessen 


RIEMANN'S  DISSERTATION.  253 

the  ninth  chapter)  that  things  and  their  properties  are 
to  be  deduced  from  our  concepts  of  them.  As  I  have 
already  said,  Riemann  does  not  define  the  term  "  con- 
cept ; "  nor  does  he  inquire  how  concepts  are  formed 
or  how  they  come  to  be  possessions  of  the  intellect. 
He  says,  indeed,  that  concepts  of  quantity  are  possible 
only  when  they  can  be  subsumed  under  higher  concepts, 

unbedenklich  von  der  Forderung  ausgehen,  gegebene  Dinge  als  gleich- 
artig  zu  betrachten),  dagegen  sind  die  Veranlassungen  zur  Bildung  von 
Bcgriffen,  deren  Bestimmungsweisen  eine  stetige  Mannigfaltigkeit  bilden, 
im  gemeinen  Leben  so  selten,  dass  die  Orte  der  Sinnengegenstaende  und 
die  Farben  wohl  die  einzigen  einfachen  Begriffe  sind,  deren  Bestimmungs- 
weisen eine  mehrfach  ausgedehnte  Mannigfaltigkeit  bilden.  Haeufigere 
Veranlassung  zur  Erzeugung  und  Ausbildung  dieser  Begrime  findet  sich 
erst  in  der  hoehern  Mathematik. 

"Bestimmte,  durch  ein  Merkmal  oder  eine  Grenze  unterschiedene 
Thtile  einer  Mannigfaltigkeit  heissen  Quanta.  Ihre  Vergleichung  der 
Quantitaet  nach  geschieht  bei  den  discreten  Groessen  durch  ZaeMung,  bei  den 
stetigen  durch  Messung.  .  .  .  Fuer  den  gegenwaertigen  Zweck  genuegt  es, 
aus  diesem  allgemeinen  Tlieile  der  Lehre  von  den  ausgedehnten  Groessen, 
wo  weiter  nichts  vorausgesetzt  wird,  als  was  in  dem  Begriffe  derselben  ent- 
halten  ist,  zwei  Puncte  hervorzuheben,  wovon  der  erste  die  Erzeugung  des 
Begriffs  einer  mehrfach  ausgedehnlen  Mannigfaltigkeit,  die  zweite  die  Zu- 
rueckfuehrung  der  Ortsbestimmungen  in  einer  gegebenen  Mannigfaltigkeit 
auf  Quantitaetsbestimmungen  betrifft,  und  das  loesentliche  Kennzeichen  einer 
n-fachen  Ausdehnung  deutlich  machen  wird.'1'' 

I  ought  to  say  that  my  interpretations  of  several  passages  of  this  text 
are  more  or  less  conjectural.  There  is  room  for  serious  doubt,  for  in- 
stance, whether  the  expression  "  Bestimmungsweisen  "  is  meant  to  denote 
the  species  comprehended  by  a  genus,  or  the  parts  constituting  a  whole. — 
A  wretched  translation  of  Riemann's  essay,  which,  by  its  clumsy  literalism, 
materially  adds  to  the  obscurity  and  confusion  of  the  original,  was  pub- 
lished in  1873,  by  W.  K.  Clifford  (Nature,  vol.  viii,  pp.  14  and  36  seq.). 
This  translation  was  no  doubt  made,  not  by,  but  for,  Professor  Clifford, 
by  some  one  who  had  a  very  insufficient  knowledge  of  German.  The 
merits  of  the  translation  are  not  unfairly  instanced  in  the  rendering  of  Rie- 
mann's term  "  Mannigfaltigkeiten  "  (varieties,  multiplicities,  used  in  the 
sense  of  multiples — Helmholtz  translates  "aggregates")  by  "manifold- 
nesses,"  of  "  Groessenbegriffe "  by  "  magnitude-notions,"  etc.  Of  one 
passage  the  whole  sense  is  changed  by  reading  koennten  for  konnten. 


254  CONCEPTS  OF  MODERN  PHYSICS. 

or,  as  he  expresses  it,  "  when  there  is  a  general  concept 
which  admits  of  different  specializations."  But  the 
question,  where  this  process  of  subsumption  begins  or 
ends,  and  what  are  the  nature  and  origin  of  the  highest 
concept  or  summum  genus  of  which  all  inferior  genera 
or  species  must  be  specializations,  does  not  occur  to  him. 
It  is,  however,  an  inevitable  conclusion  from  Riemann's 
first  proposition  itself  that  he  holds  this  most  general 
concept  to  be  an  a  priori  form  or  possession  of  the 
mind,  and  that  he  believes  the  process  of  deduction  by 
which  its  specializations  are  derived  from  it  -  to  be  (in 
the  language  of  Kant)  a  series  of  synthetic  judg- 
ments a  priori.  In  view  of  this  a  further  consideration 
of  the  proposition  is  unnecessary  ;  it  is  refuted  by  the 
whole  tenor  of  the  preceding  chapters  of  this  book. 
I  may  be  permitted  to  observe,  however,  that  it  is  with- 
out parallel  in  the  entire  history  of  intellectualism 
(usually  called  idealism) ;  Kant,  for  example,  expressly 
disclaims  all  belief  in  the  doctrine  that  the  intellect  is 
aboriginally  furnished  with  ready-made  concepts. 

2.  The  second  proposition,  that  concepts  of  quantity 
can  be  formed  and  determined  only  by  subsumption 
under  more  general  concepts,  is  probably  a  vague  remi- 
niscence of  the  old  logical  rule  that  all  definition  is  per 
genus  et  differentiam.  In  spite  of  Riemann's  com- 
plaint, in  the  second  sentence  of  his  essay,  that  hitherto 
the  science  of  geometry  has  given  nominal  definitions 
only  of  space  and  constructions  in  space — a  complaint, 
by  the  way,  which,  so  far  as  it  applies  to  constructions 
in  space,  is  unfounded — he  does  not  seem  to  have  a 
very  clear  insight  into  the  nature  of  the  distinction  be- 
tween definitions  and  concepts.  For,  if  he  had  prop- 
erly realized  this  distinction,  he  could  not  have  failed 
to  ask  himself  the  question,  what,  under  his  definition, 


RIEMANN'S  DISSERTATION.  255 

became  of  the  summum  genus  "  quantity  "  which  is 
the  logical  terminus  of  the  processes  of  subsumption 
of  which  he  speaks.  Is  this  summum  genus  also  a 
concept  ?  Then  it  must  be  subsumable,  in  conformity 
with  his' rule,  under  a  still  higher  concept,  which,  ex  vi 
termini,  it  is  not,  being  itself  the  highest.  Or  is  it 
something  else — a  datum  of  experience  ?  If  it  is,  how 
then  is  the  second  proposition  to  be  reconciled  with  the 
first,  according  to  which  everything  is  to  be  deduced 
from,  as  well  as  subsumed  under,  a  concept  ?  Or  is 
this  the  old  case  of  the  hen  in  Newmarket  which  lays 
an  egg,  from  which  the  same  hen  presently  comes  forth 
as  a  chicken-? 

The  proposition  here  discussed  almost  at  the  outset 
involves  our  author  in  the  most  intolerable  perplexity. 
"  Concepts,"  he  says,  "  whose  specializations  form  a 
discrete  aggregate  (or  multiple)  are  so  common  that,  in 
the  more  cultivated  languages  at  least,  a  concept  may 
always  be  found  under  which  things  of  whatever  kind 
are  subsumable."  The  meaning  of  this  is,  I  take  it, 
that  of  discrete  aggregates  there  are  always  several 
similar  or  connatural  kinds  or  species  which  may  read- 
ily be  subsumed  under  a  higher  concept.  "  But,"  he 
continues,  "  the  occasions  for  the  formation  of  concepts, 
whose  specializations  constitute  a  continuous  aggregate, 
are  so  rare  in  ordinary  life,  that  the  places  of  things 
and  colors  are  probably  the  only  simple  concepts  whose 
specializations  constitute  a  multiply  extended  aggre- 
gate " — that  is  to  say,  I  suppose,  there  is  but  one  spe- 
cies of  a  continuous  aggregate  or  multiple  other  than 
space  that  admits  of  coordination  and  subsumption 
with  it  under  the  concept  "  multiply  extended  aggre- 
gate," viz.,  color.  This  singular  statement  (which,  it 
may  be  noted  parenthetically,  is  the  exact  reverse  of 


256  CONCEPTS  OF  MODERN  PHYSICS. 

the  truth,  there  being,  as  we  shall  hereafter  see,  but 
one  kind  of  discrete  quantities,  viz.,  numbers,  and  in- 
numerable kinds  of  continuous  quantities)  has  been 
elaborated  with  an  extravagant  expenditure  of  analyti- 
cal power  by  Benno  Erdmann,*  who  finds  that  there 
are  two  triply  extended  multiples  which  are  coordinate 
and  subsumable  with  space  of  three  dimensions  under 
the  concept  of  a  "continuous  multiply  extended  ag- 
gregate :  "  sound  and  color.  Sound,  according  to  Erd- 
mann,  is  a  function  of  three  independent  variables, 
acuteness,  intensity,  and  timbre  (Klangfarbe).  Similarly 
color  depends  on  the  variables  tone,  degree  of  saturation 
(Saettigungsgrad),  and  intensity. f 

All  this  is  simply  puerile.  To  imagine  that  conclu- 
sions respecting  the  nature  of  space  and  the  origin  of 
its  concept  can  be  drawn  from  the  mere  fact  that  space 
is  a  function  of  three  variables,  and  may  thus  in  a 
manner  be  classified  with  similar  functions,  is  a  mock- 
ery of  all  reasoning  from  which  an  old  scholastic  would 
have  turned  with  the  scornful  reminder  that  coordina- 
tion and  subsumption,  for  the  purpose  of  effectually 
aiding  in  the  formation  of  a  particular  concept,  must 
not  only  be  under  a  genus,  but  under  a  genus  proxi- 
mum.  ;f  Weissenborn's  remark,*  that  on  the  same  log- 

*  Die  Axiome  der  Geometrie  (Leipzig,  1877)  p.  40  seq. 

f  It  is  significant,  in  this  connection,  that  according  to  Helmholtz 
(who  also  falls  in  with  Riemann's  theory  of  conception)  the  three  vari- 
ables of  the  function  "  color  "  are  the  three  primary  colors  of  which 
each  several  color  is  said  to  be  a  mixture.  "  The  Origin  and  Meaning," 
etc.  Mind,  vol.  i,  p.  309. 

%  Of  this  Erdmann  seems  to  have  some  inkling,  for  he  notes  that 
space  differs  from  color  and  sound  in  the  circumstance  of  the  absolute 
interchangeability  of  its  three  dimensions,  the  "  dimensions  "  of  color 
and  sound  not  being  interchangeable. 

*  "  Ueber  die  neueren  Ansichten  vom  Raum,"  Vierteljahrsschrift  fuer 
wissenschaftliche  Philosophie,  vol.  ii,  p.  321. 


RIEM ANN'S  DISSERTATION.  '257 

ical  principles  space  miglit  be  coordinated  with  the 
amount  of  interest  produced  by  a  certain  capital,  which 
is  a  function  of  the  three  variables  capital,  rate  of  in- 
terest, and  time,  is  perfectly  just.  And  the  number  of 
species  coordinate  with  space  in  the  same  sense  might 
be  indefinitely  increased.  For  instance,  space  might 
be  coordinated  with  the  velocity  of  a  railway-train  on 
a  straight  road,  inasmuch  as  this  velocity  is  a  function 
of  the  motive  power  of  the  engine,  the  weight  of  the 
train,  and  the  grade  of  the  track  ;  or  with  the  volatility 
of  a  liquid,  which  is  a  function  of  the  nature  of  the 
liquid,  its  temperature,  and  the  pressure  of  the  atmos- 
phere ;  or  with  the  capacity  of  a  man  for  labor,  which 
depends  on  his  general  health  and  strength,  the  quantity 
of  nourishment  he  has-  taken,  and  the  amount  of  sleep 
he  has  had ;  and  so  on  indefinitely.  All  this  is  very 
absurd,  but  not  more  so  than  the  coordination  of  space 
with  color  and  sound  on  the  mere  basis  of  the  depend- 
ence of  each  on  three  variables  which  are  arbitrarily 
called  "  dimensions." 

3.  I  come  now  to  Riemann's  third  proposition,  that 
space  is  an  "  n-fold  extended  multiple  "  or  a  "  multiply 
extended  aggregate  "  (eine  mehrfaeh  oder  n-fach  ausge- 
dehnte  Mannigfaltigkeit  ").  The  term  "  Mannigfaltig- 
keit," as  here  employed,  is  a  standing  puzzle  to  the 
readers  of  Riemann's  essay.  "Weissenborn,  who  justly 
objects  to  the  use  of  an  adjective  or  predicative  word 
in  an  appellative  sense,  for  the  denotation  of  a  substan- 
tive entity,  conjectures  *  that  it  was  expressly  devised 
by  Riemann  for  the  purpose  of  bringing  the  concept 
"  space  "  within  the  scope  of  his  second  proposition. 
But  this  is  a  mistake.  Riemann  adopted  the  term  from 
Gauss,  who  was  probably  the  originator  of  its  employ- 


258  CONCEPTS  OF  MODERN  PHYSICS. 

ment  for  the  designation  of  "  space  in  general "  (as 
distinguished  from  "  flat  space,"  in  the  metageometrical 
sense).*  Gauss,  in  turn,  took  the  expression,  no  doubt, 
from  Herbart,  f  to  whose  attempt  at  an  elaboration  of 
the  idea  of  space  from  the  manifold  qualitative  data 
of  sense  I  have  already  referred,  and  whose  philosophy 
is,  to  a  great  extent,  a  sort  of  reproduction  of  the  old 
Eleatic  quandaries  about  "  The  One  and  the  Many." 
Herbart,  in  fine,  had  obtained  it  from  Kant,  whose  dis- 
ciple he  was,  or  believed  himself  to  be,  and  whose 
phrase  "  Mannigfaltigkeiten  der  Empfindung  "  is  vari- 
ously found,  not  only  in  his  own  writings,  but  also  in 
those  of  his  followers. 

The  only  comment  which  I  deem  it  necessary  to 
make  on  this  proposition  is  that  space  is  not  a  "  multiple  " 
or  "  aggregate  "  at  all,  but  that  its  very  essence  is  con- 
tinuity. This,  as  has  been  abundantly  shown,  follows 
from  its  conceptual  nature  as  well  as  from  its  relativity. 
The  determination  of  points  in  space,  or  "  elements  " 
of  space,  results  from  the  establishment  of  quantitative 
relations  between  its  parts,  i.  e.,  its  purely  arbitrary 
divisions,  by  means  of  numbers,  in  the  manner  to  be 
considered  presently.  I  have  already  shown,  in  the 

*  In  his  Anzeige  of  the  Theoria  residuorum  biquadraticorum,  Com- 
mentatio  secunda,  Gauss  says :  "  Der  Verfasser  hat  sich  vorbehalten, 
den  Gegenstand  welcher  in  der  vorliegenden  Abhandlung  eigentlich  nur 
gelegentlich  beruehrt  ist,  kuenftig  vollstaendig  zu  bearbeiten,  wo  dann 
auch  die  Frage,  warum  die  Relationen  zwischen  Dingen^  die  eine  Mannig- 
fcdtigkeit  von  mehr  als  zwei  Dimensionen  darbieten,  nicht  noch  andere,  in 
der  allgemeinen  Arithmetic  zulaessige  Arten  von  Groessen  liefern  koennen, 
ihre  JBeantwortung  finden  wird."  Gauss,  Werke,  vol.  ii,  p.  178.  This 
notice  appeared  originally  in  the  Goettingische  Gelehrte  Anzeigen  of 
April  25,  1831. 

f  In  his  Synechologie,  e.  g.,  Herbart  speaks  of  "  die  MannigfaltigJceit 
der  irrationalen  Fortschreitungen  in  Bezug  auf  den  Raum"  Herbart's 
Werke,  vol.  iv,  p.  163. 


RIEMANN'S  DISSERTATION.  ^      259 

last  chapter,  that  ppace  itself  is  not,  in  any  intelligible 
sense,  a  quantity. 

4.  Riemann's  fourth  proposition  is  founded  on  a 
confusion  between  conceptual  possibility  and  Teal  or 
empirical  possibility.  Conceptual  possibility  is  deter- 
mined solely  by  the  consistency  or  inconsistency  of  the 
elements  of  the  concept  to  be  formed — it  is  tested  sim- 
ply by  the  logical  law  of  non-contradiction ;  while  em- 
pirical possibility  depends  upon  the  consistency  of  the 
thing  conceived  with  the  various  conditions  of  sensible 
reality,  or,  what  is  the  same  thing,  the  laws  of  nature. 
This  subject,  also,  has  already  been  discussed  to  some 
extent  in  the  last  chapter,  where  it  was  pointed  out  that 
conceivability  (in .  the  strict  sense  of  the  term)  of  a 
thing  or  phenomenon  is  no  proof  of  its  imaginability 
or  representability  under  the  conditions  of  our  physical 
and  intellectual  organization.  Upon  this  distinction 
depend  the  utility  and  scope  of  the  artifice  not  unfre- 
quently  resorted  to,  in  certain  analytical  investigations, 
of  supposing  the  existence  of  a  fourth  spatial  dimen- 
sion for  the  purpose  of  reducing  certain  functions  to  a 
symmetrical  form ;  and  this  distinction,  too,  is  the  basis 
of  an  observation  made  by  Boole*  twenty-six  years 


"  Space  is  presented  to  us,  in  perception,  as  possess- 
ing the  three  dimensions  of  length,  breadth,  and  depth. 
But  in  a  large  class  of  problems  relating  to  the  prop- 
erties of  curved  surfaces,  the  rotation  of  solid  bodies 
around  axes,  the  vibration  of  elastic  media,  etc.,  this 
limitation  appears  in  the  analytical  investigation  to  be 
of  an  arbitrary  character,  and,  if  attention  were  paid 
to  the  processes  of  solution  alone,  no  reason  could  be 
discovered  why  space  should  not  exist  in  four,  or  in  any 

*  Laws  of  Thought,  p.  175,  note. 


260  CONCEPTS  OF  MODERN  PHYSICS. 

greater  number  of,  dimensions.  The  intellectual  pro- 
cedure in  the  imaginary  world  thus  suggested  can  be 
apprehended  by  the  clearest  light  of  analogy."  Upon 
the  same  ground,  and  in  the  same  sense,  Hermann 
Grassmann,  who  is  sometimes  referred  to  as  one  of  the 
founders  of  transcendental  geometry,  has  developed 
the  theory  of  extension  in  its  general  application  to  an 
indefinite  number  of  dimensions,  although  he  certainly 
did  not  cherish  the  delusion  (as  seems  to  be  supposed 
by  Victor  Schlegel  *)  that  this  could  be  the  source  of 
inferences  respecting  the  number  of  actual  or  empiri- 
cally possible  dimensions  of  space.  On  this  subject  we 
have  Grassmann's  own  explicit  declaration :  f  "It  is 
clear,"  he  says,  "that  the  concept  of  space  can  in  no 
wise  be  generated  by  thought.  .  .  .  Whoever  maintains 
the  contrary  must  undertake  to  derive  the  dimensions 
of  space  from  the  pure  laws  of  thought — a  problem 
which  is  at  once  seen  to  be  impossible  of  solution." 

5.  Closely  akin  to  his  third  and  fourth  propositions 
is  Riemann's  fifth  proposition,  that  continuous  quanti- 
ties are  coordinate  with  discrete  quantities,  both  being 
in  their  nature  multiples  or  aggregates,  and  therefore 
species  of  the  same  genus.  This  pernicious  fallacy  is 
one  of  the  traditional  errors  current  among  mathema- 
ticians, and  has  been  prolific  of  innumerable  delusions. 
It  is  this  error  which  has  stood  in  the  way  of  the  forma- 
tion of  a  rational,  intelligible,  and  consistent  theory  of 
irrational  and  imaginary  quantities,  so  called,  and  has 
shrouded  the  true  principles  of  the  doctrine  of  "  com- 
plex numbers  "  and  of  the  calculus  of  quaternions  in  an 
impenetrable  haze. 

The  proposition  that  discrete  and  continuous  quan- 

*  System  der  Raumlehre,  preface,  p.  vi. 

t  Die  lineare  Ausdehnungslehre  (1844)  Einleitung,  p.  20  seq. 


RIEMANN'S  DISSERTATION.  261 

titles  are  coordinate  species  of  the  same  genus  amounts 
to  nothing  less  than  the  thesis  that  signs  are  logically 
coordinate  with  their  significates.  There  are  no  "  dis- 
crete quantities "  except  those  which  are  dealt  with  in 
special  (common)  and  general  arithmetic,  that  is  to  say, 
numbers.  ISTow,  a  number  is  an  aggregate  or  collection 
of  units  each  of  which  simply  represents  an  act  of  ap- 
prehension, whatever  may  be  the  extent  or  nature  of 
the  object  apprehended.  If  this  object  is  designated 
as  a  quantity,  a  number  is  not  a  quantity  at  all,  nor  a 
measure  of  quantity,  but  simply  an  intellectual  vehicle 
of  quantities — a  purely  subjective  instrumentality  for 
their  comparison  and  admeasurement.  All  the  uncer- 
tainty and  confusion  which  are  characteristic  of  the 
numerous  attempts  to  define  and  classify  quantities  are 
due  to  the  ignorance  or  neglect  of  this  elementary 
truth.  Quantity  has  been  defined  as  "that  which  is 
susceptible  of  augmentation,  diminution  and  division," 
and  as  "  the  genus  of  which  magnitude  and  multitude 
are  the  species ; "  or  quantities  have  been  first  divided 
into  extensive  quantities  (space)  and  intensive  quantities 
(forces,  colors,  sounds,  and  all  subjective  affections),  and 
the  extensive  quantities  have  then  been  subdivided  into 
continuous  and  discrete.  Now,  the  fact  is  that  all  ob- 
jects of  apprehension,  including  all  data  of  sense,  are 
in  themselves,  i.  e.,  within  the  act  of  apprehension,  es- 
sentially continuous.  They  become  discrete  only  by 
being  subjected,  arbitrarily  or  necessarily,  to  several 
acts  of  apprehension,  and  by  thus  being  severed  into 
parts,  or  coordinated  with  other  objects  similarly  appre- 
hended into  wholes.  To  say  that  a  datum  of  sensation 
or  of  subjective  feeling  is  in  itself  discrete  is  to  assert 
that  it  is  absolute,  and  to  deny  that  quantity  is  essen- 
tially relative.  And  to  maintain  (with  those  who  speak 
12 


262  CONCEPTS  OF  MODERN  PHYSICS. 

of  positive,  negative,  fractional,  irrational,  imaginary, 
complex,  linear,  or  directional  numbers)  that  number 
may  be  continuous  is  to  ignore  the  plainest  and  most 
unmistakable  fact  in  all  our  intellectual  operations,  and 
to  misinterpret  all  the  teachings  of  the  history  of  mathe- 
matics. Numbers,  in  themselves,  being  mere  groups 
or  series  of  acts  of  intellectual  apprehension  without 
reference  to  their  contents,  are  not  and  can  not  be  posi- 
tive or  negative,  much  less  fractional,  irrational,  or  im- 
aginary. They  can,  indeed,  be  applied,  not  only  to  data 
of  sensation  and  of  subjective  feeling,  but  also,  by  anal- 
ogy, to  relations  between  them,  including  relations  es- 
tablished by  the  intellect.  They  can,  therefore,  stand, 
not  only  for  things,  but  also  for  their  actions  and  reac- 
tions and  for  the  operations  to  which  they  are  subjected. 
A  number  may  represent  motion  in  a  given  direction 
and  in  the  direction  opposite  to  it,  thus  becoming  af- 
fected by  the  signs  plus  and  minus  ;  but  these  signs  do 
not  indicate  any  change  in  the  nature  of  numbers,  but 
merely  a  particularity  in  their  application.  Similarly 
numbers  may  represent  ratios  and  assume  the  form  of 
fractions ;  but  the  numbers  do  not  thereby  cease  to  be 
what  they  are,  viz.,  units  or  collections  of  units,  and 
therefore  essentially  integers.  Fractions  can  be  prop- 
erly called  numbers  only  in  the  sense  that  they  point  to 
the  division,  not  of  the  primary  units  expressive  of  the 
original  acts  of  apprehension,  but  of  ike  objects  appre- 
hended, into  subordinate  units.  Again :  numbers  may 
be  signs  of  operations  upon  quantities  that  can  not  be 
successfully  performed,  such  as  the  reduction  of  the 
diagonal  and  the  side  of  a  square  to  a  common  measure 
— in  other  words,  the  establishment  of  a  definite  numer- 
ical ratio  between  two  quantities  which  do  not  admit 
of  such  a  ratio.  In  such  case  the  futility  of  the  attempt 


RIEMANN'S  DISSERTATION.  '  263 

finds  expression  in  a  sign  prefixed  to  a  number  which, 
together  with  its  significate,  is  ordinarily  termed  an 
irrational  quantity ;  but  the  irrationality  lies,  not  in  the 
number,  but  in  the  attempt  at  its  application  to  incom- 
mensurable magnitudes.  The  same  thing  is  true,  muta- 
tis mutandis )  of  "  imaginary  quantities  "  and  "  complex 
numbers."  The  object  of  the  act  of  apprehension, 
which  is  represented  by  the  numerical  unit,  may  be, 
not  only  rectilinear  motion  or  transference  in  a  given 
direction,  but  also  angular  motion ;  as  the  calculus  of 
quaternions  expresses  it,  the  unit  of  operation  may  be  a 
vector,  or  a  versor,  or  both;  whence  it  follows  that 
whenever  the  attempt  is  made  to  represent  such  an  op- 
eration in  terms  of  linear  units  with  their  positive  or 
negative  prefixes  indicative  of  a  fixed  direction  in  which 
the  motions,  whereof  the  lines  are  the  measures,  occur, 
the  attempt  again  fails,  and  this  fact  emerges  in  the  form 
of  the  symbol  which  (being  part  of  a  system  of  symboli- 
zation  that  is  not  comprehensive  enough  to  embrace  the 
new  operation)  assumes  a  so-called  imaginary  form.  But 
here  once  more,  it  is  not  the  number  which  is  imagi- 
nary, "but  the  operation  as  interpreted  in  conformity 
with  the  conventional  rules  of  symbolization^  the  con- 
sequence being  that  these  rules  have  to  be  extended,  and 
that  the  meaning  of  the  symbols  has  to  be  widened. 
But  this  again  imports  a  change,  not  in  the  nature  of 
the  signs,  i.  e.,  of  the  numbers,  but  in  the  nature  and 
extent  of  their  significates.  In  this  manner  the  scope 
of  arithmetical  (and,  of  course,  algebraic)  symbolization 
is  continually  extended,  not  only  by  enlarging,  but  also 
by  wholly  changing  the  things,  relations  or  operations 
which  are  successively  the  objects  of  intellectual  appre- 
hension. All  this  is  perfectly  safe  and  legitimate,  pro- 
vided that  the  change  in  the  signification  of  the  symbols 


264  CONCEPTS  OF  MODERN  PHYSICS. 

be  made  in  conformity  with  the  logical  canon  of  con- 
sistency, and  with  due  regard,  moreover,  to  the  effect 
of  such  change  upon  the  validity  of  the  rules  govern- 
ing the  syntheses  and  analyses  to  which  the  symbols  are 
subjected.  In  the  operation  of  ordinary  arithmetical 
or  algebraic  multiplication,  for  instance,  the  law  of  com- 
mutation is  of  universal  validity.  Multiplication  being 
nothing  more  than  an  abbreviated  addition,  the  multi- 
plicand and  the  multiplier  may  exchange  places  or  func- 
tions without  any  effect  upon  the  result.  In  the  calcu- 
lus of  quaternions  the  mathematician  generalizes  the 
principle  of  multiplication,  defining  it  as  a  process  of 
finding  a  quantity  which  is  produced  from,  or  related 
to,  the  multiplicand  in  the  same  way  in  which  the  mul- 
tiplier is  produced  from,  or  related  to,  the  unit.  Under 
this  new  definition  he  multiplies  lines  and  other  quan- 
tities into  each  other ;  but  now  it  appears  that  the  law 
of  commutation  is  no  longer  generally  applicable.  The 
reason  is  that  the  apparent  expansion  of  the  principle 
of  multiplication  was  in  fact  also  a  limitation,  or  rather 
a  shifting  of  the  meaning  of  the  arithmetical  or  alge- 
braic symbol — a  removal  of  the  condition  upon  which 
the  validity  of  the  law  of  commutation  depended.  I 
may  observe  here,  incidentally,  that  it  is  a  mistake  to 
say,  with  Kelland  and  others,  that  the  calculus  of  quater- 
nions grows  out  of  the  common  arithmetical  or  alge- 
braic calculus  by  the  removal  of  limitations.  The 
example  just  adduced  shows  that  it  may  involve  an  im- 
position of  limitations  as  well.  For  this  reason  Pea- 
cock's law,  which  he  calls  the  "  principle  of  the  perma- 
nence of  equivalent  f  orms,*  viz.,  that "  whatever  algebrai- 
cal forms  are  equivalent,  when  the  symbols  are  general 
in  form  but  specific  in  value,  will  be  equivalent  likewise 

*  Peacock,  Symbolical  Algebra,  p.  59. 


RIEMANN'S  DISSERTATION.  265 

when  the  symbols  are  general  in  value  as  well  as  in 
forms,"  in  order  to  be  available  as  the  fundamental 
principle  of  the  theory  of  "  complex  numbers,"  requires 
a  modification  far  more  serious  than  is  implied  in  Han- 
kel's  new  statement  of  it  as  "  the  principle  of  the  per- 
manence of  formal  laws."  For  the  expression  "  formal 
laws  "  is  ambiguous  and  leaves  us  in  doubt  as  to  what 
laws  are  formal  in  the  sense  of  being  applicable  to  all 
the  operations  which  are  in  any  way  representable  by 
arithmetical  or  algebraic  symbols. 

The  error  respecting  the  true  nature  and  function  of 
arithmetical  and  algebraic  quantities  has  become  next 
to  ineradicable  by  reason  of  the  inveterate  use  of  the 
word  "quantity"  for  the  purpose  of  designating  indis- 
criminately both  extended  objects  or  forms  of  exten- 
sion and  the  abstract  numerical  units  or  aggregates  by 
means  of  which  their  metrical  relations  are  determined. 
The  effect  of  this  indiscriminate  use  is  another  illustra- 
tion of  the  well-known  fact  in  the  history  of  cognition 
that  words  react  powerfully  upon  the  thoughts  of  men, 
and  by  this  reaction  become  productive  of  incalculable 
error  and  confusion.  It  is  not  to  be  expected,  of  course, 
that  mathematicians  will  cease,  at  this  late  day,  to  speak 
of  arithmetical  or  algebraic  symbols  as  "  quantities ; " 
but  there  may  be  a  little  hope  for  the  suggestion  that 
they  might  return  to  the  old  phrase  "  geometrical  (and 
other)  magnitudes."  The  mischief  lies,  not  so  much  in 
the  use  of  a  particular  word,  as  in  the  employment  of 
the  same  word  for  the  denotation  of  objects  differing 
from  each  other  toto  genere* 

*  The  perplexities  occasioned  by  the  use  of  improper  and  misleading 
terms  in  mathematics  are  animadverted  on  by  Gauss  himself  in  the  notice 
already  cited  (Werke,  vol.  ii,  p.  178),  where  he  speaks  of  the  obscurity 
incident  to  the  interpretation  of  "  negative  and  imaginary  numbers,"  and 


266  CONCEPTS  OF  MODEKN  PHYSICS. 

The  ignorance  or  oblivion  of  the  distinction  here 
referred  to  also  illustrates  a  phase  in  the  history  of  error 
exemplifications  of  which  have  repeatedly  been  met 
with  in  the  preceding  pages  :  the  confusion  between 
purely  conventional  forms  of  thought  and  speech  and 
forms  or  laws  of  objective  existence.  This  confusion, 
which  is  at  the  bottom  of  the  old  assumption  that  our 
arbitrary  or  conventional  classifications  of  natural  phe- 
nomena are  coincident  with  essential  distinctions  be- 
tween them  and  can  be  used  as  a  source  of  inferences 
respecting  their  nature  and  origin  —  that,  as  some  one 
has  said,  the  score  of  the  Lord's  creation,  like  that  of 
Haydn's  Creation,  is  crossed  with  bars  —  has  been  pro- 
lific of  an  endless  train  of  fanciful  presumptions  by 
which  the  progress  of  science  is  incessantly  obstructed. 

For  the  reasons  here  set  forth,  the  terms  "  abstract 
and  concrete  numbers  "  are  also  fallacious  and  mislead- 
ing. Numbers,  in  themselves,  are  essentially  abstract. 
In  another  sense  they  are  necessarily  concrete  :  they  al- 
ways stand  for  some  particular  object,  relation,  or  opera- 
tion. They  are  nothing  in  themselves.  This  remark 
is  doubly  true  of  algebraic  symbols  which  require  in- 
terpretation, in  the  first  place,  by  assigning  to  them 
particular  numerical  values,  these,  in  turn,  remaining 
without  significance  until  the  units,  of  which  they  con- 
sist, are  referred  to  their  proper  objects,  relations,  or 
operations.  This  is,  no  doubt,  Duehring's  meaning 
when  he  observes,  somewhere  in  his  History  of  the 
Principles  of  Mechanics,  that  algebraic  symbolization  is 
radically  defective  inasmuch  as  it  makes  no  display  of 
the  numerical  units  which  are  the  essential  coefficients 


observes:  "If  +  1,  —  1,  \r^l  had  not  been  called  positive,  negative, 
imaginary  (or  even  impossible)  units,  but,  for  example,  direct,  inverse, 
lateral  units,  this  obscurity  would  have  vanished." 


RIEMANN'S  DISSERTATION.  267 

of  every  literal  symbol.  He  might  have  extended  this 
observation  by  adding  that  the  use  of  letters  as  alge- 
braic symbols,  i.  e.,  as  representatives  of  numbers,  is  in 
itself  a  serious  (though,  perhaps,  an  unavoidable)  in- 
firmity of  mathematical  notation.  In  the  simple  for- 
mula, for  instance,  expressive  of  the  velocity  of  a  moving 

body  in  terms  of  space  and  time  fv  =  ^  Y  the  letters  have 

a  tendency  to  suggest  to  the  mathematician  that  he  has 
before  him  direct  representatives  of  the  things  or  ele- 
ments with  which  he  deals,  and  not  merely  of  their 
ratios  expressible  in  numbers.  In  every  algebraic  opera- 
tion the  use  of  letters  obscures  the  real  nature,  both  of 
the  processes  and  of  the  results,  and  tends  to  strengthen 
ontological  prepossessions. 

The  true  theory  of  the  relations  between  arithmet- 
ical or  algebraic  quantities  and  magnitudes  of  extension 
was  stated  long  ago,  in  Germany  by  Martin  Ohm  and 
in  England  by  George  Peacock  (the  Dean  of  Ely),  Au- 
gustus de  Morgan,  D.  F.  Gregory,  and  others ;  but  the 
writings  of  these  thinkers  have  produced  little  impres- 
sion upon  contemporary  and  succeeding  generations  of 
mathematicians.  This  is  peculiarly  apparent  in  the 
books  and  articles  expository  of  the  theories  of  "im- 
aginary quantities  "  and  "  complex  numbers,"  and  of 
the  doctrines  of  the  calculus  of  quaternions.  The  im- 
mense extension  of  the  sphere  of  analysis  since  Des- 
cartes's  new  application  of  algebra  to  the  determination 
of  geometrical  magnitudes  is  almost  universally  attrib- 
uted to  a  growing  insight  into  the  true  character  of 
"  arithmetical  quantities,"  and  to  a  progressive  explica- 
tion of  the  essential  implications  of  number.  It  is  sup- 
posed that  Euclid's  denial  of  the  existence  of  numeri- 
cal ratios  between  incommensurable  quantities,  as  well 


268  CONCEPTS  OF  MODERN  PHYSICS. 

as  the  protests  of  the  early  -occidental  arithmeticians 
and  algebraists  against  negative  or  irrational  numbers  as 
"numeri  dbsurdi  infra  nil"  or  "numerificti"  or  the 
designation  by  Girolamo  Cardano  of  the  negative  roots 
of  an  equation  as  ."  cestimationes  fictce "  representing 
solutions  " vere  sophisticoe"  are  one  and  all  simply  evi- 
dences of  the  ignorance  of  these  several  writers  of  the 
real  nature  of  numbers.  It  is  not  at  all  unusual  to  meet 
with  the  dogma,  in  treatises  on  the  theory  of  "  complex 
numbers,"  that  algebra  and  arithmetic  are  essentially 
linear,  numeration  being  impossible  except  by  progres- 
sion, in  equal  steps,  in  the  direction  of  a  straight  line.* 
And,  I  may  add,  the  belief  is  by  no  means  uncommon 
that  metageometry  is  an  advance  beyond  the  old  doc- 
trines concerning  the  relations  between  geometrical 
forms  in  ordinary  space,  in  the  same  sense  and  by  the 
same  logic  in  and  by  which  the  calculus  of  quaternions  is 
an  advance  beyond  ordinary  analytical  geometry. 

The  foregoing  discussion  has  brought  us  to  the  point 
where  the  reader  is  in  a  condition,  I  hope,  to  realize  the 
great  fundamental  absurdity  of  Eiemann's  endeavor  to 
draw  inferences  respecting  the  nature  of  space  and  the 
extension  of  its  concept  from  algebraic  representations 
of  "  multiplicities."  An  algebraic  multiple  and  a  spa- 
tial magnitude  are  totally  disparate.  That  no  conclu- 
sions about  forms  of  extension  or  spatial  magnitudes  are 
derivable  from  the  forms  of  algebraic  functions  is  evi- 
dent upon  the  most  elementary  considerations.  The 
same  algebraic  formula  may  stand  for  the  most  various 
things.  Equations  of  the  second  degree,  for  example, 
may  represent  either  geometrical  areas,  or  geometrical 
curves.  The  equation  y  =  a?2  may  represent,  either  the 
area  of  a  square  whose  side  is  x,  or  a  parabola  (referred 

*  Cf.  Riecke,  die  Rechnung  mit  Richtungszahlen  (Stuttgart,  1856). 


RIEMANN'S  DISSERTATION.  269 

to  an  axis  of  ordinates)  whose  parameter  is  1.  If  Rie- 
mann's  argument  were  fundamentally  valid,  it  could  be 
presented  in  very  succinct  and  simple  form.  It  would 
be  nothing  more  than  a  suggestion  that,  because  alge- 
braic quantities  of  the  first,  second,  and  third  degrees 
denote  geometrical  magnitudes  of  one,  two,  and  three 
dimensions  respectively,  there  must  be  geometrical  mag- 
nitudes of  four,  five,  six,  etc.,  dimensions  correspond- 
ing to  algebraic  quantities  of  the  fourth,  fifth,  sixth, 
etc.,  degree.* 

It  is  hardly  necessary  to  say,  after  all  this,  that  the 
analytical  argument  in  favor  of  the  existence,  or  possi- 
bility, of  transcendental  space  is  another  flagrant  in- 
stance of  the  reification  of  concepts. 

*  It  is  not  unworthy  of  remark,  here,  that  the  practice  of  reading  a;2  and 
x3  as  x  square  and  x  cube,  instead  of  x  of  the  second  or  third  power,  is 
founded  upon  the  silent  or  express  assumption  that  an  algebraic  quantity 
has  an  inherent  geometric  import.  The  practice  is,  therefore,  mislead- 
ing, and  ought  to  be  disused.  Principiis  obsta  ! 


CHAPTER  XY. 

COSMOLOGICAL     AND     COSMOGENETIC    SPECULATIONS. THE 

NEBULAR   HYPOTHESIS. 

LIKE  all  metaphysical  theories,  the  atomo-mechanical 
theory  has  its  cosmogonies.  All  metaphysical  cosmog- 
onies are  attempts  to  deduce  the  universe  and  its  phe- 
nomena from  one  or  more  primordial  elements  by  the 
application  of  a  few  general  principles,  The  cosmogo- 
nies of  the  atomo-mechanical  theory  are  attempts  to 
deduce  the  universe  and  its  phenomena  from  the  ele- 
ments of  mass  and  motion  by  the  application  of  mechan- 
ical principles  expressive  of  the  simple  laws  of  motion. 
As  has  been  shown,  the  ultimate  problem  of  the  atomo- 
mechanical  theory,  to  whose  effectual  and  complete  so- 
lution the  physicists  of  the  day  look  forward  with  a 
greater  or  less  degree  of  confidence — though  many  of 
them  are  clear-sighted  enough  to  regard  it  as  an  aspira- 
tion never  to  be  realized — is  the  exhibition  of  all  vital 
and  organic  phenomena  as  results  of  ordinary  chemical 
and  physical  action,  and  of  chemical  and  physical  action, 
in  turn,  as  exchanges  and  transferences  of  mechanical 
motion  between  constant  and  uniform  elements  of  mass. 

A  question  necessarily  preliminary  to  cosmological 
speculations  of  whatever  kind  has  been  extensively 
mooted,  of  late,  by  mathematicians  and  physicists  alike 
— the  question  respecting  the  finitude  or  infinitude  of 


COSMOLOGICAL  SPECULATIONS.  271 

the  universe  in  time,  space,  and  mass.*  A  cosmogony, 
properly  so  called,  inevitably  involves  the  presumption 
that  the  universe  is  finite  in  past  time  at  least,  for  it  is 
a  theory  respecting  the  origin  or  beginning  of  the  uni- 
verse. The  vision  of  the  cosmogenetic  theorist  extends 
backward,  either  to  the  absolute  nothing,  or  to  a  state 
of  physical  uniformity  wholly  destitute  of  those  phe- 
nomenal differences  and  changes  which  are  the  essential 
prerequisites  of  the  notion  of  time.  This  universal 
cosmogenetic  presumption  of  the  finite  duration  of  the 
universe  in  the  past  has  recently  been  supplemented  by 
the  assertion  of  its  limited  duration  in  the  future — an 
assertion  founded  on  a  variety  of  physical  considerations, 
the  most  noteworthy  among  which  is  the  doctrine  of 
the  progressive  dissipation  of  energy.  This  doctrine  is 
stated  in  the  most  intelligible  form,  perhaps,  by  Sir 
William  Thomson,f  and  is  embodied  in  the  following 
propositions  : 

"  1.  There  is  at  present  in  the  material  world  a 
universal  tendency  to  the  dissipation  of  mechanical  en- 
ergy. 

"  2.  Any  restoration  of  mechanical  energy,  without 
more  than  an  equivalent  of  dissipation,  is  impossible  in 
inanimate  material  processes,  and  is  probably  never 
effected  by  material  masses  either  endowed  with  vege- 
table life,  or  subjected  to  the  will  of  an  animated  creat- 
ure. 

"  3.  Within  a  finite  time  past  the  earth  must  have 
been,  and  within  a  finite  period  of  time  to  come  the 
earth  must  again  be,  unfit  for  the  habitation  of  man  as 

*  Of.  Wundt,  "  Ueber  das  Kosmologische  Problem,"  Vierteljahrs- 
schrift  fuer  wissenschaftliche  Philosophic,  vol.  i,  p.  80  seq.  ' 

t  "  On  a  Universal  Tendency  in  Nature  to  the  Dissipation  of  Mechani- 
cal Energy,"  Phil.  Mag.,  series  iv,  vol.  x,  p.  304  seq. 


272  CONCEPTS  OF  MODERN  PHYSICS. 

at  present  constituted,  unless  operations  have  been,  or 
are  to  be,  performed  which  are  impossible  under  the 
laws  to  which  the  known  operations  going  on  at  present 
in  the  material  world  are  subject." 

The  reasoning  by  which  these  conclusions  (which,  it 
may  be  noted  in  passing,  are  carefully  and  in  terms 
confined  to  our  planet,  or,  at  least,  our  planetary  sys- 
tem) are  arrived  at  is  that,  inasmuch  as  all  the  opera- 
tions of  nature,  which  constitute  its  life  and  action, 
depend  upon  transformations  of  energy,  and  as  every 
such  transformation,  in  conformity  with  the  second  law 
of  thermo-dynamics,  is  in  effect  (to  use  the  expression 
of  P.  G.  Tait)  a  degradation  from  a  plane  of  higher  to 
one  of  lower  transformability  or  availability,  the  ulti- 
mate effect  must  be  a  conversion  of  all  the  energy  of 
the  world  into  heat  and  a  reduction  of  its  temperature 
to  absolute  uniformity.  From  this  state  of  uniformity 
in  the  diffusion  of  heat  no  restoration  of  available  en- 
ergy is  possible  ;  for  heat  admits  of  transformation  into 
other  forms  of  energy  only  by  passing  from  a  body  of 
higher  to  one  of  lower  temperature.* 

*  The  doctrine  of  the  dissipation  of  energy  has  been  extensively  de- 
veloped by  Clausius,  who  designates  the  sum  of  the  possible  transforma- 
tions of  the  world's  energy  as  its  entropy,  and  announces  that  "  the 
entropy  of  the  world  tends  to  a  maximum."  (Pogg.  Ann.,  vol.  cxxi,  p.  1 ; 
Abhandlungen  ueber  die  mechanische  Waermetheorie,  vol.  ii,  p.  44.)  It 
is  to  be  regretted  that  Tait,  while  adopting  the  word  "  entropy,"  under- 
takes to  use  it,  as  he  himself  says  (Thermo-dynamies,  §  48 ;  ib.,  §  178), 
"  in  the  opposite  sense  to  that  in  which  Clausius  employed  it,"  and  that 
Maxwell  (Theory  of  Heat,  pp.  186,  188)  follows  him.  Nothing  is  more 
to  be  reprobated  than  an  arbitrary  change  in  scientific  terminology,  and 
especially  a  deliberate  tampering  with  the  received  meaning  of  a  term. 
It  ought  to  be  added  that  Tait  does  not  even  succeed  in  his  attempt  to 
reverse  Clausius's  meaning,  and  that  Maxwell,  too,  is  in  error  when  he 
says  that  "  Clausius  uses  the  word  (entropy)  to  denote  the  part  of  energy 
which  is  not  available." 


COSMOLOGICAL  SPECULATIONS.  273 

It  is  clear  that,  if  the  law  of  the  dissipation  of  energy 
applies  to  the  universe  at  large — that  is  to  say,  if  the 
dynamics  of  a  finite  material  system  can  be  legitimately 
extended  to  the  Cosmos  as  an  infinite  whole — there 
must,  sooner  or  later,  be  an  end  of  the  universe  identi- 
cal with  its  beginning  as  assumed  by  the  atomo-mechan- 
ical  theory.  The  processes  of  nature  must  eventuate 
in  a  thorough  homogeneity  of  its  elements — in  a  com- 
plete absence  of  the  differences  and  changes  which 
constitute  the  attestation  of  its  real  or  actual  existence. 
This  conclusion  has  been  sought  to  be  avoided  by  the 
assumption  of  the  finitude  of  the  universe  in  mass,  or  in 
space,  or  in  both.  The  first  impulse  in  this  direction 
probably  came  from  an  article  of  W.  M.  Rankine* 
(published  shortly  after  the  appearance  of  that  of  Sir 
William  Thomson),  in  which  it  was  argued  that  "if 
there  is  between  the  atmospheres  of  the  heavenly  bodies 
an  interstellar  medium  perfectly  transparent  and  dia- 
thermanous — i.  e.,  incapable  of  converting  light  and 
heat  from  the  radiant  into  the  fixed  or  conductible 
form,  and  thus  incapable  of  acquiring  any  temperature 
whatever — and  if  this  interstellar  medium  has  bounds 
beyond  which  there  is  empty  space,  the  radiant  heat  of 
the  world  will  be  totally-reflected  and  will  ultimately 
be  reconcentrated  into  foci  in  which  a  star  (i.  e.,  an 
extinct  mass  of  inert  compounds)  would  be  vaporized 
and  resolved  into  its  elements,  a  store  of  chemical  force 
being  thus  reproduced  at  the  expense  of  a  corresponding 
amount  of  radiant  heat." 

The  supposition  of  the  finitude  of  the  mass  of  the 
universe  was  not  new  ;  it  had  often  been  made  before. 
But  here  it  presented  itself  in  a  new  form.  Hitherto 

*  "  On  the  Reconcentration  of  the  Mechanical  Energy  of  the  Uni- 
verse," Phil.  Mag.  (iv),  vol.  iv,  p.  358  seq. 


274  CONCEPTS  OF  MODERN  PHYSICS. 

the  supposition  had  been  that  the  mass,  though  limited, 
was  diffused  throughout  unlimited  space;  and  in  this 
form  it  has  recently  been  revived  by  Wundt,  who  im- 
agines that  the  finitude  of  a  mass  may  be  reconciled 
with  the  infinitude  of  its  volume  by  the  assumption  of 
an  endlessly  progressive  increase  of  its  tenuity,  the 
mass  being  taken  as  the  finite  sum  of  an  infinite  con- 
verging series.  Kankine,  on  the  contrary,  required  the 
physicist  to  grant  that  the  mass  of  the  universe  is  finite 
also  in  extent  and  is  everywhere  surrounded  by  void 
space.  The  conception  of  a  material  universe  thus 
bounded  in  boundless  space  obviously  presents  insur- 
mountable difficulties ;  and  in  view  of  these  difficulties 
many  astronomers  and  physicists  hailed  with  delight 
the  thesis  of  the  metageometers  that  space  itself,  though 
unlimited  by  reason  of  its  inherent  curvature,  is  not 
infinite,  and  that,  therefore,  the  mass  of  the  universe 
must  be  finite,  however  diffused.  This  thesis  was 
doubly  welcome  because  it  appeared,  at  first  sight, 
also  to  afford  the  means  of  escape  from  another  diffi- 
culty raised  by  the  astronomers.  In  1826  Gibers  * 
observed  that,  if  the  number  of  bodies  in  the  universe 
radiating  heat  and  light  is  infinite,  each  point  in  space 
must  receive  an  infinite  number  of  caloric  and  luminar 
rays,  and  must,  therefore,  be  infinitely  hot  and  bright- 
adding,  however,  that  this  consequence  could  be  avoid- 
ed by  supposing  an  absorption  of  the  greater  part  of 
these  rays  by  the  dark  and  cold  bodies  in  space.  But 
this  salvo  at  once  appeared  questionable,  on  the  reflec- 
tion that  the  dark  and  cold  bodies  disseminated  among 
the  luminous  stars  must  speedily  reach  the  point  of 
incandescence,  and  that  their  absorbing  power  must 
soon  be  exhausted. 

*  Bode's  astron.  Jahrbuch,  1826,  p.  110  seq.     Quoted  by  Zoellner. 


COSMOLOGICAL  SPECULATIONS.  275 

There  is  supposed  to  be  a  still  further  and  similar 
perplexity,  growing  out  of  the  fact  of  gravitation,  espe- 
cially in  view  of  its  instantaneous  action.  It  is  said 
that  a  universe  consisting  of  an  infinite  number  of 
bodies  attracting  each  other  would  not  only  be  without 
a  definite  center  of  gravity  to  which  all  cosmical  mo- 
tions could  be  referred — its  center  of  attraction  being 
everywhere,  and  therefore  nowhere — but  would  result 
in  an  infinite  pressure  (I  follow  the  expression  of 
Wundt,  though  it  would,  perhaps,  be  more  correct  to 
say  an  infinite  strain)  at  every  point  in  space.  This 
difficulty,  in  particular,  is  urged  by  Wundt  as  insuper- 
able so  long  as  the  mass  of  the  universe  is  held  to  be 
infinite ;  it  can,  in  hik  opinion,  be  overcome  only  by 
the  assumption  that  this  mass  is  limited. 

It  is  unnecessary  to  enter  upon  a  minute  examina- 
tion of  the  validity  of  these  considerations  adduced  in 
support  of  the  theory  of  the  finitude  of  the  material 
universe.  As  to  the  last  of  them,  relating  to  the  ef- 
fects of  radiation  and  gravitation,  it  is  readily  seen,  and 
has  been  pointed  out  by  Lasswitz,*  that  they  lose  their 
force  the  moment  we  recollect  that  the  intensity,  both 
of  radiation  and  gravity,  decreases  as  the  square  of  the 
distance  increases,  and  that  the  infinite  series  expressive 
of  the  several  effects  of  heat,  light,  and  gravitation  are 
converging,  their  summation  yielding  finite  results. 
And  of  the  application  of  the  doctrine  of  the  dissipa- 
tion of  energy  to  an  infinite  universe  it  is  to  be  said 
that  it  is  wholly  inadmissible.  That  doctrine  is,  no 
doubt,  irrecusable  in  its  application  to  any  finite  mate- 
rial system.  Every  such  system  must  come  to  an  end, 
as  it  has  had  a  beginning.  And  this  is  true  of  every 
such  system,  whatever  its  extent.  But  it  is  not  true  of 

*  Vierteljahrsschrift  f.  w.  P.,  vol.  i,  p.  329  seq. 


276  CONCEPTS  OF  MODERN  PHYSICS. 

a  universe  absolutely  unlimited.  Neither  the  law  of 
the  conservation  of  energy,  nor  that  of  its  dissipation, 
can  be  legitimately  applied  to  it.  The  universe,  taken 
as  absolutely  infinite,  is  not  a  conservative  system  and 
is  not  in  any  proper  sense  subject  to  physical  laws. 
We  can  not  deal  with  the  Infinite  as  with  a  physically 
real  thing,  because  definite  physical  reality  is  coexten- 
sive with  action  and  reaction ;  and  physical  laws  can 
not  be  applied  to  it,  because  they* are  determinations  of 
the  modes  of  interaction  between  distinct,  finite  bodies. 
The  universe,  so  called,  is  not  a  distinct  body,  and 
there  are  no  bodies  without  it  with  which  it  could  in- 
teract. Operations  with  the  term  Infinite  in  analogy 
to  operations  with  finite  terms  *are  as  illegitimate  in 
physics  as  they  are  in  mathematics.  The  Infinite  is 
simply  the  expression  of  the  essential  relativity  of  all 
material  things  and  their  properties,  and  is  thus,  in  a 
sense,  inherent  in  every  finite  form.  It  is  the  basis  of 
all  the  relations  which  constitute  sensible  actuality,  but 
it  is  not  itself  a  group  of  such  relations.  It  is  the  back- 
ground of  all  material  actions  and  forms ;  no  system  of 
elements  or  forces  can  exist  without  it,  or  is  cognizable 
without  reference  to  it ;  and  in  this  sense,  and  in  this 
sense  only,  the  universe  is  necessarily  infinite  in  mass  as 
well  as  in  space  and  in  time. 

It  follows  that  all  cosmogonies  wh'ich  purport  to  be 
theories  of  the  origin  of  the  universe  as  an  absolute 
whole,  in  the  light  of  physical  or  dynamical  laws,  are 
fundamentally  absurd.  The  only  question  to  which  a 
series  or  group  of  phenomena  gives  legitimate  rise  relates 
to  their  filiation  and  interdependence ;  and  the  attempts 
to  transcend  the  bases  of  this  filiation  and  interdepend- 
ence— to  determine  the  conditions  of  the  emergence  of 
physical  phenomena  beyond  the  bounds  of  space  and 


COSMOLOGICAL  SPECULATIONS.  277 

the  limits  of  time — are  as  futile  as  (to  use  the  happy 
simile  of  Sir  William  Hamilton)  the  attempt  of  the 
eagle  to  outsoar  the  atmosphere  in  which  he  floats  and 
by  which  alone  he  may  be  supported. 

This  leads  me  to  a  discussion  of  a  cosmogenetic  the- 
ory which  has  attained  to  great  celebrity  and  very  gen- 
eral acceptance,  under  the  name  of  the  Nebular  Hy- 
pothesis. As  now  generally  held,  this  theory  may  be 
briefly  stated  as  follows : 

Primordially  the  materials,  which  are  at  present 
found,  partly  at  least,  conglomerated  in  the  bodies  com- 
posing the  stellar,  solar,  planetary,  satellitic,  and  me- 
teoric systems,  were  uniformly  dispersed  throughout 
space.  In  some  way;  by  the  action  of  cosmic  (attrac- 
tive and  other)  forces,  this  uniformly  diffused  and  very 
attenuated  matter  came  to  be  divided  into  large  nebu- 
lous spheres  which  began  slowly  to  rotate,  the  rotation 
resulting,  perhaps,  from  the  act  of  division,  or  from  in- 
ternal differences  in  their  densities  and  irregularities  in 
their  forms,  which  deflected  the  lines  of  gravitation 
from  a  strictly  radial  direction,  the  centers  of  attraction 
no  longer  coinciding  with  the  centers  of  figure.  In 
proportion  as  these  spheres  parted  with  their  heat  they 
contracted ;  and  this  contraction  led  to  an  increase  of 
their  velocities  of  rotation  in  conformity  to  a  mechani- 
cal law  known  as  the  law  of  the  conservation  of  areas 
or  of  angular  momentum  This  law,  in  its  most  gen- 
eral expression,  is  simply  a  corollary  from  the  law  of 
inertia,  from  which  it  follows  that  the  resultant  angular 
momentum  of  any  material  system  can  not  be  changed, 
either  in  magnitude  or  the  direction  of  its  axis,  by  the 
mutual  action  of  its  constituents.*  For  the  purpose  of 

*  All  mechanical  or  dynamical  laws  of  conservation — the  conserva- 
tion of  momentum,  of  angular  momentum  and  of  energy — are  (as  I  have 


278  CONCEPTS  OF  MODERN  PHYSICS. 

its  application  to  a  rotating  nebulous  mass,  however,  the 
law  may  be  more  intelligibly  stated  in  another  form, 
viz.,  that,  whatever  change  of  volume  or  form  may  be 
produced  in  a  material  system  by  the  mutual  attraction 
of  its  constituent  elements,  the  sum  of  all  the  areas  de- 
scribed by  the  radii  vectores  of  the  several  elements  or 
particles  round  the  center  of  rotation,  in  a  unit  of  time, 
is  constant.  Now,  the  areas  being  proportional  to  the 
squares  of  the  diameters,  it  follows  that  the  angular 
velocity  increased  with  great  rapidity  as  the  contraction 
of  a  nebulous  mass  proceeded.  An  immediate  conse- 
quence of  this  increase  of  velocity  was  a  proportionate 
increase  of  the  centrifugal  force  in  the  equatorial  regions 
of  the  rotating  sphere,  so  that  in  course  of  time  this 
force  came  to  balance,  and  afterward  to  exceed,  the 
centripetal  gravitation.  This  led  at  first  to  a  dispro- 
portionate contraction  of  the  sphere  at  the  poles  and  to 
the  assumption,  by  the  sphere,  of  an  oblately  spheroidal 
or  lenticular  form,  and  eventually  to  successive  detach- 
ments of  equatorial  rings  or  zones  which  at  first  circu- 
lated round  the  residual  mass  in  the  direction  of  its 
original  rotation,  but  which — by  reason  of  the  instabili- 
ty of  such  rings  in  case  of  the  least  departure  from  ab- 
solute regularity  of  form  or  constitution — broke  up  into 
parts,  forming  one  or  more  minor  spheres  or  spheroids. 
These  continued  to  revolve  round  the  sun  with  a  veloci- 
ty nearly  equal  to  the  rotatory  velocity  of  their  mate- 
already  indicated  in  the  sixth  chapter)  at  bottom  nothing  more  than  ap- 
plications of  the  principle  of  inertia  to  complex  material  systems.  It  is 
the  great  merit  of  Poinsot  to  have  brought  to  light  the  formal  analogies 
(prefigured,  to  a  certain  extent,  in  the  writings  of  Euler)  between  the  laws 
governing  movements  of  rotation  and  those  determining  the  forms  of 
ordinary  translatory  motion.  It  is  hardly  necessary  to  add  that  the  law 
of  the  conservation  of  areas  is  in  form  a  generalization  of  Kepler's  second 
law. 


COSMOLOGICAL  SPECULATIONS.  279 

rials  at  the  moment  of  their  detachment  and  congloba- 
tion.  In  most  cases,  probably,  the  whole  mass  of  such 
a  ring  coalesced  into  a  single  body,  i.  e.,  into  a  planet, 
while  in  some  cases  several  bodies  were  formed,  such  as 
they  appear  in  our  planetary  system  in  the  zone  of 
asteroids.  Each  of  the  planets,  while  revolving  round 
the  residual  mass  whose  condensation  is  supposed  to 
have  produced  the  sun,  also  began  to  rotate  on  an  axis 
of  its  own,  the  direction  of  this  rotation  coinciding  with 
that  of  its  revolution.  It  thus  became  subject  to  the 
same  dynamical  conditions  which  determined  the  evolu- 
tion of  the  parent  system ;  it  also  threw  off  rings  which 
either  retained  their  form  (as  in  the  case  of  the  Satur- 
nian  rings)  or  formed  into  minor  satellitic  bodies. 

The  arguments  which  have  been  advanced  in  sup- 
port of  this  hypothesis  are  so  well  known  that  it  is 
hardly  necessary  to  recapitulate  them.  Among  them 
are  the  existence,  in  the  stellar  regions,  of  nebulous 
masses  in  various  stages  of  condensation  ;  the  evidences 
of  the  increase  of  temperature  from  the  surface  of  our 
planet  toward  the  interior ;  the  proximate  coincidence 
of  the  orbital  motions  of  the  several  planets,  both  in 
direction  and  plane,  and  the  further  proximate  coinci- 
dence of  this  orbital  motion  with  the  direction  and 
plane  of  the  sun's  rotation ;  the  similar  coincidence  of 
the  directions  of  the  orbital  motions  of  the  satellites 
with  the  axial  motions  of  their  planets ;  the  oblately 
spheroidal  form  of  the  earth,  and,  as  far  as  we  know,  of 
the  other  planets,  which  peculiar  form  has  not  only 
been  theoretically  demonstrated,  but  has  also  been  ex- 
perimentally shown,  by  M.  Plateau,  to  be  the  form 
necessarily  assumed  by  a  rotating  body  in  a  liquid  or 
semi-liquid  state.  These  considerations  were  adduced,  al- 
most in  the  same  order  and  form,  by  Kant  and  Laplace, 


280  CONCEPTS  OF  MODERN  PHYSICS. 

and  they  have  since  been  supplemented  by  a  variety  of 
other  considerations  more  or  less  plausible,  among  which 
may  be  mentioned  the  agreement  of  the  theoretical  con- 
sequences of  the  fact,  that  the  projection  of  planetary 
masses  from  the  parent  globe  must  have  taken  place 
with  ever-increasing  rapidity  as  the  contraction  of  the 
globe  progressed,  with  certain  well-known  features  of 
our  own  planetary  system.  .Attempts  not  wholly  un- 
successful have  even  been  made  to  effect  a  deduction, 
from  the  elements  of  this  theory,  of  the  empirical  law 
respecting  the  distances  of  the  several  planets  from  the 
sun  which  is  known  as  the  law  of  Bode  or  Titius. 

The  nebular  hypothesis,  as  a  theory  of  the  origin, 
not  only  of  our  planetary  system,  but  of  stellar  and 
planetary  systems  throughout  the  universe,  is  common- 
ly ascribed  to  Laplace,  who  is  supposed  to  have  been 
unaware  of  the  fact  that  the  hypothesis  which  he  ad- 
vanced had  been  published  by  Kant,  in  his  Naturge- 
schichte  des  Himmels,  m  1?55,  nearly  half  a  century 
before  the  first  appearance  of  the  Exposition  du  Sys- 
teme  du  Monde,  in  1796.  But  the  truth  is  that  the 
Nebular  Hypothesis,  in  the  form  in  which  it  is  now 
generally  held,  is  due  to  Kant,  and  differs  in  several 
essential  particulars  from  the  hypothesis  of  Laplace. 
This  latter  hypothesis  is  limited  in  terms  to  our  plane- 
tary system,  and  there  is  no  indication  in  any  of  the 
writings  of  the  French  astronomer — certainly  none  in 
his  Exposition  du  Systeme  du  Monde— thai  he  ventured 
to  extend  it  to  the  entire  universe,  as  was  expressly 
done  by  Kant.  But  there  is  a  difference  still  more  im- 
portant between  the  hypotheses  of  the  two  thinkers. 
Kant's  assumption  was  that  "  all  the  materials  compos- 
ing the  spheres  that  belong  to  our  solar  worlcf  were,  in 
the  beginning  of  all  things,  resolved  into  their  element- 


COSMOLOGICAL  SPECULATIONS.  281 

ary  substance  and  filled  the  whole  space  of  the  system 
in  which  these  spheres  now  move."  *  This  assumption 
is  common  to  all  recent  forms  of  the  nebular  hypoth- 
esis that  have  fallen  under  my  notice — they  all  postu- 
late a  diffusion  of  the  entire  mass  of  the  sun,  planets, 
comets,  and  satellites  constituting  our  planetary  system 
throughout  the  planetary  space.  The  assumption  of 
Laplace,  on  the  contrary,  is  simply  that  tl\e  atmosphere 
of  the  sun  at  one  time  extended  beyond  the  orbits  of 
the  farthest  planets,  and  that  the  formation  of  the 
planets  and  their  satellites  a.s  well  as  that  of  the  comets 
was  due  to  a  gradual  cooling  and  contraction  of  this 
atmosphere.f 

It  is  hardly  necessary  to  say  that  the  Laplacean  form 
of  the  nebular  hypothesis  is  far  too  narrow  to  serve  the 
purposes  of  a  general  cosmological  theory.  Such  a 
theory  demands  the  derivation  of  the  several  concre- 
tions of  cosmical  matter  from  some  primitive  homo- 
geneous mass.  This  demand  is  complied  with  by  the 
hypothesis  of  Kant ;  but .  it  is  very  partially,  if  at  all, 
satisfied  by  that  of  Laplace.  And  this  brings  us  into 
the  presence  of  a  formidable  difficulty.  It  is  to  be 
feared  that,  in  proportion  to  its  amplification  to  cosmo- 
genetic  dimensions,  the  nebular  hypothesis  parts  with 
its  validity  as  a  physical  theory.  This  subject  was  ex- 

*  "  Ich  nehme  an,  dass  alle  Materie,  daraus  die  Kugeln  die  zu  unserer 
Sonnenwelt  gehoeren,  alle  Planeten  und  Kometen  bestehen,  im  Anfang  aller 
Dinge  in  ihren  elementarischen  Grundstoff  aufgeloes't,  den  ganzen  Raum 
des  Weltgebaendes  erfuellt  haben,  darin  jetzt  diese  gebildeten  Koerper 
herumlaufen."  "  Naturgeschichte  des  Himmels,"  Kant's  Werke,  vol.  vi, 
p.  95. 

f  "  La  consideration  des  mouvemens  planetaires  nous  conduit  dpnc  a 
penser  qu'en  vertu  d'une  chaleur  excessive  ^ atmosphere  du  soldi  s'est 
primitivement  etendue  au  dela  des  orbes  de  toutes  les  planetes,  et 
qu'elle  s'est  resserre'e  successivement  jusqu'a  ses  limites  actuelles."  Sys- 
teme  du  Monde  (2me  ed.),  p.  345. 


282  CONCEPTS  OF  MODERN  PHYSICS. 

amined,  nearly  twenty  years  ago,  by  M.  Babinet,  in  an 
article  on  the  Cosmogony  of  Laplace,*  in  which  he 
shows  that  the  actual  rotatory  velocities  of  the  several 
planets  are  in  fact  vastly  greater  than  the  velocities  to 
be  deduced,  by  the  aid  of  the  law  of  the  conservation 
of  areas,  from  the  nebular  hypothesis,  if  that  hypothesis 
includes  the  assumption  of  a  diffusion  of  the  solar  mass 
itself  throughout  a  space  coextensive  with  the  limits 
of  our  planetary  system.  "  Several  persons,"  says  M. 
Babinet,  "  have  thought  that  the  sun  himself  had  origi- 
nally been  expanded  so  as  to  fill  the  entire  space  now 
occupied  by  the  planets,  although  Laplace  expressly 
mentions  that  at  the  moment  of  the  formation  of  these 
bodies  it  was  only  the  atmosphere  of  the  sun  which  had 
this  vast  extent.  We  are  able  to  test  this  question 
mathematically,  by  calculating  from  the  sun's  actual  pe- 
riod of  rotation,  which  is  twenty-five  and  three  tenths 
days,  what  would  be  the  velocity  of  rotation  if,  con- 
serving the  sum  of  the  areas  described  by  all  its  mate- 
rial points,  it  were  expanded  so  that  its  radius,  which  is 
now  equal  to  one  hundred  and  twelve  times  the  equa- 
torial radius  of  the  earth,  became  equal  to  the  distance 
from  the  earth  to  the  sun,  or  from  Neptune  to  the  sun. 
.  .  .  The  calculation  on  the  first  of  these  bases  gives 
a  rotation  of  1,162,000  days,  amounting  to  more  than 
three  thousand  (3,181)  years.  The  period  of  revo- 
lution calculated  on  the  second  basis  would  evidently 
be  nine  hundred  times  greater,  that  is  to  say,  more  than 
twenty-seven  thousand  centuries. 

"  Note  sur  un  Point  de  la  Cosmogonie  de  Laplace,"  Comptes  Rendus, 
vol.  lii,  p.  481  seq.  My  attention  was  drawn  to  this  article  by  a  passage 
in  an  interesting  little  pamphlet  of  Dr.  E.  Budde,  of  Bonn,  Zur  Kos- 
mologie  der  Gegenwart  (Bonn,  ed.  Weber,  1872),  to  which  I  shall  have 
occasion  to  recur  hereafter. 


COSMOLOGICAL  SPECULATIONS.  283 

"  These  numbers  being  infinitely  greater  than  those 
expressive  of  the  earth's  and  Neptune's  actual  periods 
of  revolution,  it  is  plainly  impossible  to  admit  that  these 
planets  have  been  formed  out  of  the  solar  mass  itself 
extended  beyond  the  planetary  orbits.  This,  however, 
does  not  preclude  the  idea  that  the  stars  themselves 
have  been  formed  at  the  expense  of  a  universal  cosmic 
matter  endowed  with  excessively  feeble  movements  of 
rotation  round  the  center  of  gravity  of  each  mass  which 
was  in  process  of  formation  as  an  independent  sun. 

"  The  conclusion  is  that,  if  the  entire  mass  of  the 
sun  had  been  expanded  to  the  limits  of  the  planetary 
system,  it  must  have  had  a  movement  of  rotation  far 
too  feeble  to  enable  the  centrifugal  force  to  balance  the 
force  of  gravity  so  as  to  lead  to  the  separation  of  an 
equatorial  ring  from  the  total  mass." 

The  discrepancies  here  brought  to  light  between  the 
actual  orbital  periods  of  the  planets  and  the  correspond- 
ing periods  found  by  calculation  in  accordance  with  the 
postulates  of  the  nebular  hypothesis,  are  so  enormous 
that  there  appears  to  be  no  possibility  of  accounting  for 
them  by  the  assumption  of  a  progressive  contraction  of 
the  orbits  of  the  several  planets  since  their  projection, 
and  the  consequent  quickening  of  their  orbital  motions. 

The  calculations  of  M.  Babinet  do  not  constitute 
the  only  difficulty  which  besets  the  nebular  hypothesis, 
either  in  its  general  cosmogenetic  or  in  its  special  La- 
placean  form.  In  the  progress  of  astronomical  dis- 
covery it  has  appeared  that  several  of  the  supposed 
coincidences  between  the  facts  and  the  hypothesis  fail. 
Thus,  there  appears  to 'be  an  exception  to  the  direc- 
tional uniformity  of  the  axial  and  orbital  motions  of 
the  planets  and  their  satellites  in  the  case  of  Uranus, 
the  orbital  planes  of  whose  satellites  are  nearly  perpen- 


284:  CONCEPTS  OF  MODERN  PHYSICS. 

dicular  to  the  ecliptic,  the  circumplanetarj  motions  of 
the  satellite  as  well  as  the  axial  motion  of  the  planet, 
moreover,  being  retrograde — a  fact  long  since  discovered 
by  Sir  William  Herschel,  and  confirmed  by  various  sub- 
sequent observations.  But  the  most  serious  blow  which 
has  lately  been  dealt  to  the  nebular  hypothesis  consists 
in  the  recent  discovery  (1877),  by  Professor  Asaph  Hall, 
of  two  satellites  of  the  planet  Mars  and  the  proximate 
determination  of  their  respective  distances  from  the 
primary  as  well  as  their  orbital  (circumplanetary)  pe- 
riods. It  was  found  that  the  distances  of  the  inner 
and  outer  satellites  from  the  center  of  the  planet  are 
about  three  and  six  times,  respectively,  the  radius  of 
the  planet,  and  that  the  periods  of  revolution  of  these 
satellites  are  7*65  and  30'25  hours,  respectively,  while 
the  period  of  rotation  of  the  planet  (Mars)  itself  is 
24*623  hours.  It  appeared,  then,  that  on-e  of  the  satel- 
lites revolves  about  the  planet  in  less  than  one  third  of 
the  twie  required  for  the  planers  axial  rotation. 

The  radical  inconsistency  of  this  fact  with  the  nebu- 
lar hypothesis  is  undeniable.  In  the  light  of  the  hy- 
pothesis in  question,  the  orbital  motions  of  a  satellite  are 
continuations  of  the  axial  motions  of  the  materials  out 
of  which  the  satellites  are  formed;  its  orbital  period 
ought,  therefore, 'to  be  equal,  proximately  at  least,  to 
the  period  in  which  the  planet  rotated  at  the  time  of 
the  satellite's  formation.  And  that  period  is  of  neces- 
sity greater  than  the  period  of  the  planet's  present  ro- 
tation, by  reason  of  the  acceleration  produced  by  its 
subsequent  contraction. 

The  attempts  thus  far  made  to  reconcile  the  anomaly 
here  referred  to  with  the  essential  postulates  of  the 
nebular  hypothesis  have  been  entirely  fruitless.  These 
attempts  are  founded  on  two  suppositions,  the  first 


COSMOLOGICAL  SPECULATIONS.  285 

being  that  the  planet's  period  of  rotation  has  been 
retarded  by  tidal  action,  and  the  second  that  the  orbits 
of  the  satellites  have  been  contracted  and  their  orbital 
periods  accelerated  by  the  resistance  of  the  aethereal  me- 
dium which  was  formerly  supposed  to  have  shortened 
the  period  of  Encke's  comet.  But  the  first  of  these  sup- 
positions, as  Professor  John  Le  Conte  has  observed,* 
is  unavailable  for  the  purpose  of  reducing  the  anomaly, 
inasmuch  as  tidal  retardation  could  at  most  produce  a 
coincidence  of  the  period  of  the  planet's  rotation  with 
the  orbital  period  of  the  satellite,  irrespective  of  the 
fact  that  the  anomaly  itself — the  continual  advance  of 
the  inner  satellite  beyond  any  given  point  of  the  planet, 
or,  in  other  words,  the  incessant  drag  exerted  by  the 
satellite  on  the  planet  in  the  direction  of  its  rotation — 
produces  a  form  of  tidal  action  which  tends  to  acceler- 
ate, instead  of  retarding,  the  rotation  of  the  planet. 
And  the  second  supposition  is,  to  say  the  least,  insuf- 
ficient to  account  for  the  anomaly,  even  if  the  -very 
doubtful  existence  of  an  interstellar  and  interplanetary 
medium  capable  of  offering  material  resistance  to  plane- 
tary motion  be  granted.  Besides,  it  is  to  be  borne  in 
mind  that  the  contraction  of  a  satellitic  orbit,  in  conse- 
quence of  the  resistance  of  the  medium  in  which  the 
satellite  moves,  does  not  entail  an  acceleration  of  its 
revolution  to  the  same  extent  to  which  such  accelera- 
tion would  be  produced  under  the  simple  action  of 
gravitative  forces,  one  of  the  concurrent,  and  indeed 
primary,  effects  of  the  resistance  being  a  retardation  of 
the  revolutionary  motion  itself. 

To  these  several  objections  to  the  nebular  hypothesis 
as  a  physical  theory  of  the  formation  and  constitution 
of  the  universe  must  be  added,  of  course,  the  funda- 

*  "Mars  and  his  Satellites,"  Popular  Science  Monthly,  November,  1879. 
18 


CONCEPTS  OF  MODERN  PHYSICS. 

mental  inadmissibility,  already  pointed  out,  of  all  specu- 
lations respecting  the  origin  of  the '  universe  as  an  un- 
limited whole.  But,  apart  from  this,  it  is  plain  that 
the  derivation  of  the  forms  and  movements  of  the 
stellar  and  planetary  systems  from  a  primordial  homo- 
geneous mass  uniformly  diffused  throughout  space  is 
impossible.  In  the  first  place,  such  a  mass  must  be 
either  at  rest  or  in  uniform  motion ;  and  this  state  of 
rest  or  uniform  motion,  according  to  the  most  element- 
ary principles,  could  be  changed  only  by  extraneous 
impulses  or  attractions.  And,  there  being  no  "with- 
out "  to  the  all-embracing  Cosmos  or  Chaos,  the  original 
state  of  rest  or  uniform  motion  would  necessarily  be 
perpetual.*  In  the  second  place,  such  a  nebulous 
universe  would  be  of  perfectly  uniform  temperature ; 
all  parts  would  be  equally  hot  (or  cold),  and  there  could 
be  no  radiation  or  loss  of  heat  resulting  in  a  contraction 
of  any  part  of  the  nebulous  mass.  Its  thermo-dynami- 
cal  condition  would  be  constant  for  the  same  reason 
which  establishes  the  permanence  of  its  general  dy- 
namical condition. 

The  cumulation  of  difficulties  presented  by  the 
nebular  hypothesis  has  become  so  great,  and  is  begin- 
ning to.be  so  extensively  realized,  as  to  develop  a  ten- 
dency to  modify  or  supplant  it  by  another  hypothesis 
which  may  be  called  the  hypothesis  of  meteoric  agglom- 
eration. This  hypothesis  commends  itself  to  the  modern 
physicist  by  reason  of  its  apparent  exemplification  of 
the  general  doctrine  that,  for  the  purpose  of  ascer- 
taining the  nature  of  the  agencies  which  have  produced 

*  As  Duehring  expresses  it  (Kritische  Geschichte  der  allgemeinen 
Principlen  der  Mechanik,  2d  ed.,  §  151),  "If  ever  there  had  been  perfect 
equilibrium  between  the  parts  (of  the  nebulous  mass)  it  would  continue 
to  exist  now." 


COSMOLOGICAL  SPECULATIONS.  287 

a  particular  physical  system  or  form,  we  must  in  the 
first  instance  look  to  the  agencies  concerned  in  its 
maintenance  or  destruction — a  doctrine  which  might 
be  condensed  into  a  canon :  quod  sustinet  vel  delet^for- 
mavit.  This  doctrine  is  in  effect  nothing  more  than  a 
new  statement  of  the  old  law  of  parsimony  which  for- 
bids the  unnecessary  multiplication  of  explanatory  ele- 
ments and  agencies.  It  has  been  extensively  and  suc- 
cessfully applied  in  geology,  which  now  endeavors  to 
account  for  all  the  past  phases  in  the  history  of  the 
earth  by  the  regular  and  ordinary  action  of  the  forces 
known  to  be  at  work  in  maintaining  or  modifying  its 
present  condition.  The  theory  of  meteoric  agglomera- 
tion was  first  suggested  by  Julius  Robert  Mayer,*  and 
was  founded  on  the  reflection  that  the  great  annual  fall 
of  meteoric  masses  upon  the  earth  indicates  the  circula- 
tion or  movement  within  our  planetary  space  of  a  vast 
number  of  small  bodies  which  must  strike  large  bodies, 
like  the  sun,  in  numbers  enormously  exceeding  those 
reaching  the  earth,  the  number  being  greater  in  pro- 
portion both  to  the  masses  and  the  surfaces  of  the 
larger  bodies.  These  meteors,  according  to  Mayer, 
are  in  a  sense  the  fuel  of  the  sun,  and  all  bodies  within 
the  planetary  system  are  subject  to  accretions,  both 
of  mass  and  temperature,  in  consequence  of  their  col- 
lisions with  them.  Now,  it  is  supposed  that  in  astro- 
nomically primeval  times  the  proportion  of  these  me- 
teoric masses  to  the  masses  of  the  large  solar  and  plan- 
etary bodies  may  have  been  far  greater  than  it  is  now 
— that,  in  fact,  there  may  have  been  a  time  when  the 
space  now  occupied  by  our  planetary  system  presented 
the  appearance  of  a  swarm  of  such  meteors  of  all  sizes 

*  In  his  Beitraege  zur  Mechanic  des  Himmels  (first  published  in  1848), 
Mechanik  der  Waerme,  p.  157  seq. 


288  CONCEPTS  OF  MODERN  PHYSICS. 

and  of  all  degrees  and  forms  of  consistency  and  aggre- 
gation, moving  about  at  all  rates  of  velocity,  in  all  di- 
rections, and  in  orbits  of  every  degree  of  eccentricity. 
These  masses  would  be  consolidated,  and  movements, 
both  of  rotation  and  revolution,  would  be  generated  in 
the  bodies  so  formed  by  their  collisions. 

At  this  point  the  question  obtrudes  itself :  how  can 
a  theory,  which  seeks  to  derive  the  orderly,  symmetrical, 
and  harmonious  world  as  we  know  it  from  the  wildest 
congestion  of  aboriginal  differences  and  anomalies — 
from  a  spring-head  of  utter  incongruity  and  confusion 
—be  made  to  account  for  the  regularities  and  coinci- 
dences whose  simple  and  natural  explanation  was  the 
conspicuous  merit  of  the  hypothesis  of  Laplace  ? 

An  answer  to  this  question  is  sought,  by  the  advo- 
cates of  the  new  theory,  in  an  appeal  to  a  principle  long 
since  established  by  Laplace  himself.  This  principle 
relates  to  the  fact  that,  amid  all  the  disturbances  caused 
by  the  mutual  attractions  of  the  planetary  bodies,  there 
exists  an  invariable  plane  passing  through  the  center  of 
gravity  of  the  whole  system,  about  which  these  bodies 
perpetually  oscillate  with  but  slight  deviations  on  either 
side.  If  on  this  invariable  plane  we  project  the  areas 
described  by  the  radii  vectores  of  the  several  elements 
of  mass  in  a  given  time,  and  multiply  each  mass  into  its 
respective  area  thus  projected,  the  sum  of  the  products 
is  a  maximum,  and  the  rate  of  its  increase  is  constant.* 
Such  a  plane  exists,  not  only  for  the  solar  system,  but 
for  any  system  of  bodies  controlled  solely  by  their  mut- 
ual attractions.  Now,  it  is  evident  that  both  the  sum 
and  the  rate  of  its  increase,  of  the  products  of  the  masses 

*  Cf.  Laplace,  Mecanique  Ce"leste,  lere  partie,  liv.  ii,  chap.  vii.  ('*  JDes 
inegalites  seculaires  des  mouvemetis  celestes.")  The  theory  was  first  pub- 
lished in  the  Journal  de  PEcole  Polytechnique,  1798. 


COSMOLOGICAL  SPECULATIONS.  289 

into  the  projections  of  the  areas  described  by  their  radii 
vectores,  are  always  less  than  the  sum  and  the  rate  of  its 
increase  of  the  products  of  the  masses  into  the  radii 
vectores  themselves,  inasmuch  as  these  radii  (unless  they 
are  parallel  to  the  plane)  are  shortened  by  their  projec- 
tion ;  and  the  difference  between  these  two  sums  is  in 
direct  proportion  to  the  deviations  of  the  movements 
from  the  direction  of  the  total  increase,  which  direction, 
for  purposes  of  reference,  is  taken  as  positive,  the  oppo- 
site direction  being,  of  course,  taken  as  negative.  And 
whenever  the  several  movements  meet  with  resistance, 
some  of  the  components  of  the  velocities  of  the  moving 
masses  are  necessarily  destroyed,  so  that  the  difference 
in  question  is  diminished  and  eventually  annulled. 
When  this  has  happened,  the  absolute  values  of  the  areas 
described  by  the  radii  vectores  of  the  masses  in  a  given 
time  become  equal  to  their  maximum  projections ;  in 
other  words,  their  planes  coincide  with  or  become  par- 
allel to  Laplace's  invariable  plane.  From  this  follows 
the  general  principle  that  the  movements  of  the  bodies 
constituting  any  finite  system,  whatever  be  their  origi- 
nal divergence  of  direction,  tend  (except  in  a  very  few 
special  cases),  by  reason  of  any  resistance  to  these  move- 
ments, to  become  parallel  to  or  coincident  with  an  inva- 
riable plane.* 

Before  leaving  this  subject  I  may  observe  that  the 
principle  just  stated,  which  admits  of  a  further  generali- 
zation, so  as  to  assume  this  form — that  all  movements  of 
the  elements  of  a  finite  material  system  depending  upon 

*  The  possible  exceptions  to  this  law  are,  of  course,  those  cases  in 
which  the  components  destroyed  are  exactly  equal  and  opposite.  The 
improbability  of  the  occurrence  of  such  cases  is  so  great  that  Budde,  who 
states  the  law  substantially  as  I  have  stated  it  in  the  text  (1.  c.,  p.  30), 
does  not  even  allude  to  the  possibility  of  an  exception. 


290  CONCEPTS  OF  MODERN  PHYSICS. 

the  mutual  action  of  such  elements  tend,  in  consequence 
of  any  permanent  interference  with  or  determination  of 
these  movements  from  without,  from  irregularity  and 
disorder  to  regularity  and  order — is,  in  my  judgment, 
one  of  the  most  important  in  the  whole  range  of  theo- 
retical physics.  For  the  condition  here  assigned — that 
the  internal  movements  of  the  system  be  subject  to 
constant  interference  from  without — is  in  fact  insepa- 
rable from  every  material  system,  there  being  no  such 
system  which  is  at  any  time  under  the  exclusive  con- 
trol of  its  own  internal  forces.  There  is,  consequently, 
in  every  finite  part  of  the  world  an  ingenerate  bias  from 
irregularity  to  regularity,  a  natural  bent  from  disorder 
to  order,  an  inherent  tendency  from  Chaos  to  Cosmos  ; 
and  this  tendency  is  the  simple  and  direct  consequence 
of  the  relativity  of  all  material  forms — of  the  fact  that 
each  finite  whole  is  always  a  part  of  a  still  greater 
whole — in  short,  that  the  finite  exists  only  on  an  ever- 
receding  background  of  infinitude.  It  is  possible  even 
that  this  principle  is  more  than  coextensive  with  the 
sphere  of  physics,  and  that,  to  a  certain  extent,  it  may 
have  its  applications  within  the  domains  of  those  sci- 
ences which  are  ordinarily  designated  as  historical. 
Although  attempts  at  a  transference  of  laws  governing 
the  interdependence  of  phenomena  whose  lines  of  con- 
nection are  simple  and  easily  traced  (such  as  the  move- 
ments of  inorganic  masses)  to  a  class  of  phenomena 
whose  relations  are  complicated  and  imperfectly  under- 
stood (such  as  the  phenomena  of  organic  and  vital  ac- 
tion) are  perilous  in  the  extreme,  and  never  to  be  made 
without  a  careful  reference  to  the  nature  and  ground  of 
the  analogies  by  which  they  are  induced,  it  is  neverthe- 
less true  that  a  great  part  of  the  progress  which  is  now 
being  made  in  the  several  departments  of  science  is 


COSMOLOGICAL  SPECULATIONS.  '     291 

due  to  liberal  exchanges,  not  only  of  results,  but  also 
of  principles  and  methods.* 

The  theory  of  meteoric  aggregation  undertakes  to 
grapple  with  still  further  elements  of  the  general  prob- 
lem of  explaining  the  actual  features  of  our  planetary 
system,  as,  for  instance,  the  comparative  inferiority  of 
the  sizes  of  the  planets  nearest  the  sun.  The  reasoning 
is  something  like  this  :  Somewhere  within  the  space 
comprising  the  various  movements  of  the  bodies,  whose 
materials  are  in  process  of  agglomeration,  a  mass  will 
probably  be  formed  which  is  preeminent  above  all  the 
others.  This — the  nucleus  of  the  future  sun  of  the 
system — must  gradually  draw  to  its  neighborhood  the 
perihelia  of  all  the  moving  meteoric  masses  or  groups. 
In  this  region,  therefore,  the  movements  of  all  the 
bodies  must  have  the  greatest  velocity ;  here  the  me- 
teors must  fly  past  each  other  with  the  greatest  swift- 
ness, and  their  approach  and  agglomeration  must  be 
most  difficult — a  circumstance  which  also  prevents  the 
rapid  growth  of  bodies  in  this  region  after  their  incho- 
ate formation.  Near  the  confines  of  the  system,  on  the 
contrary,  where  the  movements  of  the  meteors  are  slug- 
gish, the  conditions  for  the  congestion  of  large  masses 
are  comparatively  favorable.  Similarly,  a  rough  account 
is  given  of  the  fact  that  the  densities  of  the  planets  are 

*  Instances  of  the  application  of  dynamical  and,  generally,  of  physi- 
cal laws,  not  only  to  vital,  but  also  to  psychological,  action  are  afforded 
by  the  recent  discussion,  by  Avenarius,  of  the  evolution  of  thought  in 
conformity  to  the  principle  of  least  action  (Die  Philosophic  als  Denken  der 
Welt  gemaess  dem  Princip  des  Kleinsten  Kraftmaasses,  Leipzig,  1876), 
and  the  previous  discussion,  by  Schleicher,  of  the  evolution  of  language 
in  the  light  of  the  doctrine  of  natural  selection — which,  it  may  be  said 
parenthetically,  is  not  without  analogy  to  the  principle  discussed  in  the 
text — (Die  Darwin'sche  Theorie  und  die  Sprachwissenschaft,  Weimar, 
1863). 


292  CONCEPTS  OF  MODERN  PHYSICS. 

generally  in  the  inverse  ratio  to  their  sizes.  A  larger 
body  attracts  a  meteor  with  greater  intensity  than  a 
smaller  one ;  its  growth  is,  therefore,  marked  by  more 
violent  collisions  productive  of  a  higher  temperature 
and  a  corresponding  expansion. 

It  is  not  my  purpose  to  discuss  the  merits  of  this 
theory  in  detail,  or  to  express  an  opinion  as  to  its 
soundness  and  sufficiency ;  but  it  is  proper  to  say  that 
it  appears  to  me  to  stand  in  favorable  contrast  to  the 
nebular  hypothesis  precisely  by  reason  of  the  absence 
of  some  of  the  characteristics  to  which  the  general 
plausibility  of  this  latter  hypothesis  is  due.  The  nebu- 
lar hypothesis  found  ready  and  almost  enthusiastic  ac- 
ceptance, not  so  much  on  physical  as  on  metaphysical 
grounds.  The  proneness  to  derive  the  Multiple  from 
the  absolutely  Simple,  the  Various  from  the  absolutely 
Uniform,  has  its  root  in  the  second  of  the  great  structu- 
ral fallacies  which  I  have  discussed  in  the  ninth  chap- 
ter— in  the  assumption  that  the  abstract  result  of  a  gen- 
eralization, i.  e.,  a  general  concept,  may  be  made  available 
as  a  starting-point  for  the  evolution  of  the  particular 
things  subsumed  under  it.  (The  enthusiasm  for  the 
nebular  hypothesis  was,  in  this  respect,' an  ontological 
survival.  And  in  another  respect  it  was  even  more 
than  that — it  was  a  recrement  of  ancient  traditions 
about  the  origin  of  the  universe  from  Nothing.  The 
original  mist  of  the  nebular  hypothesis  is  assumed  to 
be  of  extreme  tenuity — of  a  density  less  than  the  one 
hundred  thousandth  part  of  hydrogen,  the  lightest  gase- 
ous body  known  to  the  chemist.  By  reason  of  this 
aethereal  subtilty  it  was  readily  substituted,  in  the  con- 
ceptions of  the  popular  mind,  for  the  old  void  from 
which  the  world  was  said  to  have  emerged,  and,  in  the 
imaginations  of  those  who  look  upon  matter  as  a  sort 


COSMOLOGICAL  SPECULATIONS.  293 

of  inspissation  of  Mind  for  the  universal  antemundane 
impersonal  Spirit.  It  thus  conformed  to  the  assump- 
tion that,  on  any  hypothesis  respecting  the  mode  of  the 
world's  formation,  it  must  "  in  the  beginning "  have 
been  "  without  form  and  void,"  and  at  the  same  time 
satisfied  the  mystic  yearnings  after  the  Ethereal  and 
"  Spiritualistic,"  which  is  the  special  distinction  of  that 
large  class  of  philosophers  whose  philosophy  begins 
where  clear  thinking  endsT? 


CHAPTEK  XYL 

CONCLUSION. 

1  THE  considerations  presented  in  the  preceding  pages 
lead  to  the  conclusion  that  the  atomo-mechanical  theory- 
is  not,  and  can  not  be,  the  true  basis  of  modern  physics. 
On  proper  examination,  this  theory  appears  to  be  not 
only,  as  is  generally  conceded,  incompetent  to  account 
for  the  phenomena  of  organic  life,  but  it  proves  to  be 
equally  incompetent  to  serve  as  an  explanation  of  the 
most  ordinary  cases  of  inorganic  physical  action.  And 
the  claim  that,  in  contradistinction  to  metaphysical  the- 
ories, it  resorts  to  no  assumptions,  and  operates  with  no 
elements  save  the  data  of  sensible  experience,  is  found  to 
be  wholly  inadmissible.  In  announcing  this  conclusion 
it  is  necessary,  however,  to  guard  against  two  fundamen- 
tal misconceptions.  In  the  first  place,  the  denial  of  the 
theory  of  the  atomic  constitution  of  matter,  as  it  is  gener- 
ally held  by  physicists  and  chemists,  involves  no  assertion 
respecting  the  real  constitution  of  bodies — of  chemical 
elements  or  compounds — and  certainly  does  not  imply 
the  metaphysical  thesis  of  the  absolute  continuity  of 
matter.  What  is  the  actual  constitution  of  particular 
bodies  is  a  question  to  be  determined  in  each  case  by 
experiment  and  observation.  There  is,  no  doubt,  a 
large  class  of  bodies  whose  constitution  is  molecular; 
but  from  this  it  does  not  follow  that  the  molecules 


CONCLUSION.  295 

composing  them  are  primordial,  unchangeable  units, 
existing  independently  and  in  advance  of  all  physical 
action,  and  therefore  absolutely  exempt  from  change. 
On  empirical  grounds  the  inference,  from  the  molecular 
structure  of  a  body,  of  the  permanent  existence  of  ab- 
solutely immutable  and  indestructible  atoms  or  mole- 
cules is  as  irrational  as  would  be  the  assertion  that 
primordially,  and  in  advance  of  the  formation  of  or- 
ganic bodies,  there  existed  an  indefinite  number  of 
elementary  cells,  because  all  organic  bodies  are  of  cellu- 
lar structure. 

In  the  second  place,  dissent  from  the  proposition 
that  all  physical  action  is  mechanical  in  the  sense  of 
being  a  transference  of  motion  between  distinct  masses 
by  collision  or  impact  is  not  to  be  construed  as  a  doubt 
respecting  the  constancy  of  physical  laws  or  the  univer- 
sality of  their  application.  "What  is  denied  is,  not  the 
general  dominance  of  the  law  of  physical  causation,  but 
the  doctrine  that  the  only  form  of  such  causation  is  the 
transference  of  motion  by  the  impact  of  masses  which, 
in  themselves,  are  absolutely  inert.  If  physical  action 
in  conformity  with  constant  and  uniform  law  is  des- 
ignated as  mechanical,  then  all  physical  action  is  un- 
doubtedly mechanical. 

It  may  be  said  that  physical  action  is  utterly  indeter- 
minable except  on  the  supposition  of  the  atomic  or  mo- 
lecular constitution  of  matter.  This  is  true  only  in  the 
sense  that  we  are  unable  to  deal  With  forms  of  physical 
action  otherwise  than  by  considering  them  as  modes  of 
interaction  between  distinct  physical  terms.  Physical 
action  can  not  be  subjected  to  quantitative  determination 
without  a  logical  insulation  of  the  conceptual  elements 
of  matter,  and  without  ultimate  reference  to  conceptual 
constants  of  mass  and  energy.  \  All  discursive  reasoning 


296  CONCEPTS  OF  MODERN  PHYSICS. 

depends  upon  the  formation  of  concepts,  upon  the  intel- 
lectual segregation  and  grouping  of  attributes — in  other 
words,  upon  the  consideration  of  phenomena  under  par- 
ticular aspects.  In  this  sense  the  steps  to  scientific  as 
well  as  other  knowledge  consist  in  a  series  of  logical  fic- 
tions which  are  as  legitimate  as  they  are  indispensable  in 
the  operations  of  thought,  but  whose  relations  to  the 
phenomena  whereof  they  are  the  partial  and  not  unfre- 
quently  merely  symbolical  representations  must  never 
be  lost  sight  o/^  "When  the  old  Greek  sought  to  deter- 
mine the  properties  of  the  circle,  he  began- by  con- 
structing a  polygon  whose  sides  he  subdivided  until 
they  were  supposed  to  become  infinitely  small ;  and  in 
his  view  every  line  of  definite  extent  and  form — i.  e., 
every  line  which  could  become  the  subject  of  mathe- 
matical investigation — was  composed  of  an  infinite  num- 
ber of  infinitely  small  straight  lines.  But,  he  speedily 
found  that,  while  this  fiction  enabled  him  to  deduce 
a  rule  for  calculating  the  area  of  the  circle  and  other- 
wise to  determine  a  number  of  its  properties,  never- 
theless the  circle  and  its  rectilinear  diameter  were  fun- 
damentally incommensurable,  and  the  quadrature  of 
the  circle  was  impossible.  The  modern  analyst  simi- 
larly determines  the  locus  of  a  curve  by  the  relation  of 
small  increments  of  coordinates  arbitrarily  established  ; 
but  he  is  well  aware  that  the  curve  itself  has  nothing 
to  do  with  this  arbitrary  representation,  and  he  very 
emphatically  asserts  the  continuity  of  the  curve  by  dif- 
ferentiating, or  passing  to  the  limit  of,  his  increments 
— at  the  same  time  transforming  his  coordinates  by 
changing  their  origin  or  their  inclination,  or  even  their 
system,  from  bilinears  to  polars,  whenever  he  finds  it 
convenient,  without  dreaming  that  thereby  he  is  in  the 
least  affecting  the  nature  of  the  curve  whose  properties 


CONCLUSION.  297 

are  under  discussion.  The  astronomer,  in  calculating 
the  attraction  of  a  homogeneous  sphere  upon  a  material 
point,  begins  by  assuming  the  atomic  or  molecular  con- 
stitution of  the  attracting  sphere,  establishing  a  series  of 
finite  differences  as  one  of  the  terms  of  his  equation ; 
but  thereupon  he  takes  the  series  to  be  infinite  and  the 
differences  to  be  infinitely  small,  and  very  effectually  dis- 
mantles the  molecular  scaffolding  by  integrating  instead 
of  effecting  a  summation  of  a  series  of  finite  differences. 
Observe :  the  astronomer  begins  with  two  fictions — the 
fiction  of  a  "  material  point "  (which  is,  in  truth,  a 
contradiction  in  terms),  so  as  to  insulate  the  attractive 
force  and  treat  it  as  proceeding  from  the  sphere  alone, 
and  the  fiction  of  the  finite  differences  representing  the 
molecular  constitution  of  the  sphere;  but  the  validity 
of  his  result  depends  upon  the  eventual  rescission  of 
these  fictions  and  the  rehabilitation  of  the  fact.  In 
like  manner  the  chemist  represents  the  proportions  of 
weight,  in  which  substances  combine,  as  atoms  of  defi- 
nite weight,  and  the  resulting  compounds  as  definite 
groups  of  such  atoms ;  and  this  mythical  coinage  has 
been  serviceable  in  many  ways.  But,  apart  from  the 
circumstance  that  the  symbols  have  become  wholly  in- 
adequate to  the  proper  representation  of  the  facts,  it  is 
important  to  bear  in  mind  always  that  the  symbol  is 
not  the  fact.  Newton  derived  many  of  the  leading 
optical  laws  from  his  corpuscular  theory  of  light  and 
from  the  hypothesis  of  "  fits  of  easy  transmission  and 
reflection."  His  theory  for  a  time  served  a  good  pur- 
pose ;  but  it  proved,  after  all,  to  be  but  a  convenient 
mode  of  symbolizing  the  phenomena  with  which  he 
was  familiar,  and  had  to  be  discarded  when  the  phe- 
nomenon of  interference  was  observed.  In  1824  Sady 
Carnot  deduced  the  law  of  thermic  action  which  still 


298  CONCEPTS  OF  MODERN  PHYSICS. 

bears  his  name  from  an  hypothesis  respecting  the  nat- 
ure of  heat  (supposed  by  him,  as  by  nearly  all  the  phys- 
icists of  his  time,  to  be  imponderable  matter),  which 
is  now  known,  or  universally  believed,  to  be  erroneous. 
For  certain  purposes,  such  as  the  mathematical  deter- 
mination of  gaseous  pressure  and  expansion,  thermic 
phenomena  find  a  convenient  representation  in  the  hy- 
pothesis that  a  gaseous  body  is  a  group  of  atoms  or 
molecules  in  a  state  of  incessant  motion.  Some  of  the 
properties  of  gases  have  been  successfully  deduced,  by 
Clausius  and  others,  from  formulae  founded  upon  this 
hypothesis,  and  Maxwell  has  even  succeeded  in  predict- 
ing the  phenomenon  of  the  gradual  cessation  of  the 
oscillatory  movement  of  a  disk,  suspended  between  two 
other  disks,  in  consequence  of  the  friction  of  a  gaseous 
medium,  whatever  be  the  degree  of  its  tenuity,  and 
this  prediction  has  since  been  experimentally  verified  ; 
but  neither  Clausius'3  formulae  nor  Maxwell's  experi- 
ments are  conclusive  as  to  the  real  nature  of  a  gas. 
That  no  valid  inference  respecting  the  real  constitution 
of  bodies  and  the  true  nature  of  physical  action  can  be 
drawn  from  the  forms  in  which  it  is  found  necessary  or 
convenient  to  represent  or  to  conceive  them,  is  illus- 
trated by  the  fact  that  we  habitually  resort,  not  only  in 
ordinary  thought  and  speech,  but  also  for  purposes  of 
scientific  discussion,  to  modes  of  representing  natural 
phenomena  which  are  founded  upon  views  and  hy- 
potheses long  since  discarded  as  untenable.  Just  as  we 
think  and  speak  familiarly  of  the  motions  of  the  sun 
and  stars  in  terms  of  the  old  geocentric  doctrine,  al- 
though no  one  in  our  day  doubts  the  truth  of  the  he- 
liocentric theory,  so  also  the  modern  astronomer  would 
find  it  difficult  to  dispense  with  geocentric  fictions  in 
subjecting  these  motions  to  mathematical  computation. 


CONCLUSION.  299 

Even  the  old  epicycles  survive  in  some  of  the  analyti- 
cal formulae,  by  means  of  which  that  computation  is 
effected. 

The  progress  of  modern  theoretical  physics  consists 
in  the  gradual  reduction  of  the  various  forms  of  physi- 
cal action  to  the  principle  of  the  conservation  of  energy. 
For  purposes  of  didactic  exposition  of  this  principle  we 
resort  to  the  fiction  of  systems  of  molecules  or  particles, 
whose  motions  are  simple  functions  of  the  distances  be- 
tween them.  But,  as  we  have  seen,  the  conflict  of  this 
fiction  with  the  facts  of  experience  emerges  at  once, 
when  we  undertake  to  establish  an  absolute  disjunction 
between  the  molecules  and  their  motions.  The  conser- 
vation of  energy  would  be  impossible,  if  the  ultimate 
constituents  of  a  material  system  were  in  themselves 
absolutely  inert.  And  the  same  thing  is  strikingly  ex- 
hibited in  the  recent  attempts  to  extend  the  principle  of 
the  conservation  of  energy  to  the  phenomena  of  chemi- 
cal action.  These  attempts  have  been  prompted  by  the 
observation  that  all  chemical  action  depends  upon,  or, 
at  least,  is  attended  with,  the  absorption  or  liberation 
of  heat,  and  that  the  amount  of  heat  absorbed  or  liber- 
ated is  the  measure  of  such  action.  The  determination 
of  chemical  phenomena  by  means  of  their  thermic  inci- 
dents, which  has  until  recently  been  known  as  thermo- 
chemistry, and  has  been  treated  as  a  comparatively  in- 
significant part  of  chemical  science,  is  now  coming  to 
be  regarded  as  the  true  basis  of  theoretical  chemistry. 
The  principles  of  this  new  science  have  already  been 
systematized,  to  some  extent,  in  several  distinct  trea- 
tises, among  which  may  be  mentioned  Mohr's  "Me- 
chanical Theory  of  Chemical  Affinity,"*  JSTaumann's 

*Friedrich  Mohr,  Mechanische  Theorie  der  chemischen  Affinitaet, 
Braunschweig,  1868. 


300  CONCEPTS  OF  MODERN  PHYSICS. 

" Thermo-Chemistry,"  *  and  Berthelot's  "Chemical 
Mechanics  founded  on  Thermo-Chemistry."  f 

The  importance  of  the  part  which  heat  performs  in 
chemical  transformations  was  first  distinctly  realized 
upon  the  announcement,  by  Dulong  and  Petit  in  1819, 
of  the  empirical  law,  that  the  specific  heats  of  the  ele- 
ments are  inversely  proportional  to  their  atomic  weights, 
or,  as  it  is  commonly  expressed  in  the  language  of  the 
atomic  theory,  that  the  atoms  of  all  elementary  bodies 
have  the  same  specific  heat.  Although  there  are  ap- 
parent exceptions  to  this  law  (as  in  the  cases  of  carbon, 
boron,  and  silicon),  it  holds  good  in  so  many  cases  that 
there  is  hope  of  an  explanation  of  these  exceptions  on 
grounds  on  which  they  will  utimately  prove  to  be  con- 
firmations of  the  law ;  indeed,  some  progress  in  this  di- 
rection has  already  been  made.  And  Neumann,  Ee- 
gnault,  and  Kopp  have  shown  that  the  law  applies  not 
only  to  elements,  but  also  to  compounds,  it  appearing 
that  the  specific  heat  of  a  compound  is  the  sum  of  the 
specific  heats  of  its  component  elements. 

Dulong  and  Petit's  law,  if  it  were  universally  valid, 
would  lead  to  a  remarkable  law  of  chemical  combina- 
tion. For,  it  is  obviously  identical  with  the  proposition, 
that  chemical  elements  combine  only  in  so  far  as  they 
experience  the  same  elevation  of  temperature  in  the  act 
of  combination.  It  is  not  improbable  that,  if  the  true 
relation  of  the  temperature  of  a  body  to  its  total  physi- 
cal and  chemical  energy  were  thoroughly  understood, 
this  law  would  become  one  of  the  cardinal  principles 
of  theoretical  chemistry. 

*  Dr.  Alexander  Naumann,  Grundriss  der  Thermochemie,  Braun- 
schweig, 1869. 

f  M.  Berthelot,  Essai  de  Mecanique  Chimique  fondee  sur  la  Thermo- 
chimie,  Paris,  1879. 


CONCLUSION.  301 

The  next  noteworthy  result  of  thermo-chemical  re- 
search was  the  discovery  that  the  nature  of  the  chemi- 
cal reactions  between  different  substances  depends  upon 
the  relations  between  the  specific  energies  of  the  re- 
agents as  determined  by  the  quantities  of  heat  evolved 
or  involved  in  the  progress  of  these  reactions.  It  was 
found  that  there  are  certain  elements — oxygen  and 
hydrogen,  for  example — which  combine  readily,  and, 
under  proper  conditions,  spontaneously,  the  combina- 
tion (as  Berthelot  expresses  it)  taking  place  directly, 
without  the  aid  of  extrinsic  energy,  and  being  at- 
tended with  the  evolution  of  light,  or  heat,  or  both. 
Such  combinations  are  termed  by  M.  Berthelot,  exo- 
thermic. They  result  in  the  formation  of  compounds, 
which  can  not  be  resolved  again  into  their  original 
elements  without  a  restoration  of  the  amount  of  en- 
ergy lost  in  the  combination.  On  the  other  hand, 
there  are  cases  of  endothermie  combination  in  which 
conversely  the  composition  of  the  elements  is  attended 
with  an  absorption,  and  the  decomposition  of  the  result- 
ing compound  with  a  liberation,  of  heat.  The  com- 
bination of  carbon  and  sulphur,  for  instance,  is  endo- 
thermie. Carbonic  disulphide  is  formed  by  passing 
vaporous  sulphur  over  red-hot  charcoal ;  the  union  of 
carbon  and  sulphur  is  possible  only  on  condition  of  the 
continuous  supply,  during  the  progress  of  the  union,  of 
heat,  which  is  given  out  again  when  the  disulphide  is 
resolved  into  its  elements.  The  facts  here  referred  to 
are  explained,  by  the  modern  chemist,  on  the  theory 
that  chemical  affinity  is  transformed  heat,  both  heat  and 
affinity  being  forms  of  energy ;  that  in  the  cases  of  ex- 
othermic combination  the  sum  of  the  specific  energies 
of  the  component  elements  exceeds  the  specific  energy 
of  the  compound  formed,  while  in  endothermie  com- 


302  CONCEPTS  OF  MODERN  PHYSICS. 

binations  the  specific  energy  of  the  compound  is  greater 
than  the  aggregated  specific  energies  of  the  compo- 
nents. And  it  has  been  shown  that,  whenever  we  trace 
a  number  of  elements  or  compounds  through  a  series  of 
chemical  reactions,  the  total  amount  of  energy  (appear- 
ing, before  absorption  or  after  liberation,  in  the  form  of 
heat),  which  is  liberated  or  absorbed,  is  exactly  equal  to 
the  difference  between  the  specific  energies  of  the  initial 
and  those  of  the  terminal  compounds  or  elements.  It 
is  to  be  observed  that  this  rule  applies,  not  only  to  cases 
of  composition  and  decomposition,  so  called,  but  like- 
wise to  cases  of  allotropy  and  polymerism,  inasmuch  as 
allotropic  forms  of  elements  and  isomeric  forms  of  com- 
pounds are  found  to  be  convertible  into  each  other  by 
the  addition  or  withdrawal  of  definite  amounts  of  heat. 
A  third  result  of  the  study  of  the  thermic  condition 
of  elements  and  compounds  is  the  establishment  of  the 
remarkable  principle  that  the  passage  of  any  body  or 
system  of  bodies  from  a  condition  of  a  lesser  to  one  of 
greater  stability  is  always  attended  with  evolution  of 
heat,  "  whether "  (in  the  language  of  Odling)  "  such 
change  be  what  is  commonly  called  combination,  or 
what  is  called  decomposition " ;  and  that  all  chemical 
action  which  takes  place  without  the  intervention  of 
extrinsic  energy  tends  to  the  production  of  a  body  or 
bodies  whose  formation  liberates  the  largest  amount  of 
heat.* 

*  A  sort  of  anticipation  of  this  principle  is  found  in  one  of  the  well- 
known  laws  announced  in  the  early  part  of  this  century  by  M.  Berthollet, 
in  his  "  Statique  Chimique  " — in  the  law  that,  whenever  two  soluble  salts 
are  mixed  in  solution,  they  decompose  each  other,  if  the  resulting  com- 
pound, or  mixture  of  compounds,  is  insoluble  or  less  soluble  than  the  salts 
mixed.  The  bearing  of  this  law  upon  the  principle,  stated  in  the  text,  of 
the  maximum  evolution  of  heat,  will  be  understood  upon  reference  to  the 
fact  that,  generally  speaking,  the  solubility  of  substances  is  increased  by 


CONCLUSION.  303 

This  brief  outline  sufficiently  indicates  the  facts  and 
generalizations  upon  which  it  is  proposed  to  found  the 
new  theory  of  u  Chemical  Mechanics."  Little  use  has, 
thus  far,  been  made  of  the  law  of  Dulong  and  Petit; 
but  the  other  results  of  experimental  induction  in  the 
field  of  thermo-chemistry  are  summarized  by  M.  Ber- 
thelot  in  the  introduction  to  his  work,*  as  follows : 

"  1.  Principle  of  Molecular  Work. — The  quantity 
of  heat  disengaged  in  any  reaction  whatever  is  a  meas- 
ure of  the  amount  of  chemical  and  physical  work  per- 
formed in  such  reaction. 

"  2.  Principle  of  the  Calorific  Equivalence  of  Chem- 
ical Transformations. — If  a  system  of  simple  or  com- 
pound bodies,  taken  under  determinate  conditions,  un- 
dergoes physical  or  chemical  changes  capable  of  bring- 
ing it  to  a  new  state,  without  producing  any  mechanical 
effect  outside  of  the  system,  the  amount  of  heat  liber- 
ated or  absorbed  by  the  effect  of  these  changes  depends 
solely  on  the  initial  and  final  states  of  the  system ;  it  is 
the  same,  whatever  be  the  nature  and  sequence  of  the 
intermediate  states. 

"  3.  Principle  of  Maximum  Work. — All  chemical 
change  effected  without  the  intervention  of  extraneous 
energy  tends  to  the  production  of  that  body  or  sys- 
tem of  bodies  which  liberates  the  largest  amount  of 
heat." 

This  third  principle,  as  Berthelot  observes,  may  also 
be  stated  in  the  form  that  "  all  chemical  reaction  sus- 
ceptible of  being  effected  without  the  concurrence  of 
preliminary  work  and  without  the  intervention  of  ex- 

the  application  of  heat.    Berthollet's  law,  however,  is  subject  to  excep- 
tions ;  there  are  cases  in  which  soluble  bases  are  replaced  by  insoluble 
bases,  the  result,  nevertheless,  being  the  formation  of  soluble  salts. 
*  "  M6canique  Chimique,"  pp.  xxviii,  xxix. 


304  CONCEPTS  OF  MODERN  PHYSICS. 

trinsic  energy  will  necessarily  take  place  whenever  it 
leads  to  the  evolution  of  heat." 

The  relation  of  these  propositions  to  the  doctrine 
of  the  conservation  of  energy  is  apparent.  They  are 
obviously  applications,  to  the  phenomena  of  chemical 
transformation,  of  the  two  leading  principles  which 
that  doctrine  embraces,  the  first  and  second  propositions 
of  Berthelot  representing  the  principle  of  the  correla- 
tion, equivalence,  and  mutual  convertibility  of  the  sev- 
eral forms  of  energy,  and  the  third  that  of  the  tendency 
of  all  energy  to  dissipation. 

The  study  of  chemical  changes  in  the  light  of  the 
doctrine  of  the  conservation  of  energy  exhibits  these 
changes  under  an  entirely  new  aspect.  It  shows  that 
the  question  as  to  the  possibility  of  a  chemical  "  com- 
position," or  "decomposition,"  is  as  much  a  question 
of  the  definite  proportionality  of  energies  as  of  the 
definite  proportionality  of  masses;  that  each  element 
as  well  as  each  compound  embodies  a  distinct  and 
invariable  amount  of  energy  as  well  as  a  distinct  and 
invariable  quantity  of  "  matter "  (i.  e.,  mass),  and  that 
this  energy  is  as  constitutive,  and  as  essential  a  part, 
of  the  existence  of  such  element  or  compound  as  its 
weight. 

And  here  the  question  arises:  How  is  all  this  to 
be  interpreted,  by  the  aid  of  the  ordinary  laws  of  mo- 
tion and  of  mechanical  principles  generally,  in  con- 
formity with  the  assumption  that  all  the  phenomena  of 
chemical  transformation  are  reducible  to  motions  of 
absolutely  inert  atoms  or  elements  of  mass  ?  For  that 
is  the  assumption  which  lies  at  the  base  of  the  new 
theory  of  chemical  mechanics.  Naumann  declares  in 
express  terms,  both  in  one  of  the  first  and  in  the  very 
last  of  the  sentences  of  his  book  that  "  chemistry  in  its 


CONCLUSION.  305 

ultimate  form  must  be  atomic  mechanics."  *  And  Ber- 
thelot,  though  he  avoids  the  use  of  the  word  atoms,  no 
less  explicitly  asserts  that  two  data  suffice  to  explain  the 
multiformity  of  chemical  substances  :  the  masses  of  the 
elementary  particles  and  the  nature  of  their  motion,  f 

The  explanation  of  chemical  phenomena  by  the 
theory  of  chemical  mechanics  is  to  be  effected,  then, 
by  reducing  them  to  terms  of  mass  and  motion.  On 
what  mechanical  principles  is  this  reduction  possible? 
The  fundamental  fact  to  be  accounted  for  is  the  con- 
version of  heat  into  chemical  energy.  But  this  con- 
version implies,  not  only  a  change  of  one  kind  of 
motion  into  another,  but  also  a  confinement  of  a  defi- 
nite amount  of  this  motion  to  or  within  a  definite  mass. 
According  to  the  mechanical  theory,  heat,  in  the  form  at 
least  in  which  it  is  generally  supplied  to  gaseous  bodies 
in  process  of  chemical  transformation,  consists  in  recti- 
linear atomic  or  molecular  motions  of  all  conceivable 
velocities  and  directions.  The  extent  of  these  motions 
is  limited  solely  by  the  encounters  of  the  moving 
masses.  By  these  encounters  the  range,  the  velocity, 
and  the  direction  of  the  excursion  of  every  atom  or 
molecule  are  incessantly  changed.  And,  whatever  may 
be  the  nature  of  that  form  of  motion  which  we  call 
chemical  energy,  we  know  at  least  that  a  definite  and 
invariable  amount  of  it  belongs  to  a  definite  mass  or 

*  "  Die  Chemie  in  der  fuer  sie  zu  erstrebendcn  Gestaltung  muss  sein 
eine  Mechanik  der  Atome,"  Thermochcmie,  p.  150. 

f  "  La  mature  multiforme  dont  la  chimie  etudie  la  diversite  obeit  aux 
lois  d'une  mecanique  commune.  .  .  .  Au  point  de  vuc  mecanique,  deux 
donnees  fondamentales  caracterisent  cette  diversite  en  apparence  indejinie 
des  substances  chimiques,  savoir :  la  masse  dcs  particules  elementaires,  c'est- 
d-dire  leur  equivalent,  et  la  nature  de  leurs  mouvements.  La  connaissance 
de  ces  deux  donnees  doit  suffir  pour  tout  expliquer."  Mecanique  Chi- 
mique,  tome  ii,  p.  757. 


306  CONCEPTS  OF  MODERN  PHYSICS. 

number  of  atoms  of  any  given  substance.  Whenever 
heat,  therefore,  is  converted  into  chemical  energy,  the 
motion  above  described  must,  of  necessity,  be  so  modi- 
fied that  a  definite  amount  of  it  is  brought  into  some 
sort  of  synthesis  or  union  with  a  definite  number  of 
particles.  But  that  is  certainly  impossible  if  the  par- 
ticles are  mere  inert  masses,  whose  motions  are  deter- 
mined solely  by  the  impact  of  other  masses,  as  the 
mechanical  theory  assumes.  The  specialization  or  ni- 
di vidualizat  ion  of  motion,  which  is  required,  can  be 
accounted  for  in  no  other  way  than  by  attributing  to 
the  masses  themselves  some  inherent  coercive  power. 
Even  if  an  individualization  of  heat-movements  could 
result  mechanically  from  the  collision  of  inert  particles 
—by  the  conversion  of  rectilinear  into  rotatory  motion, 
for  instance,  as  a  consequence  of  oblique  impacts — there 
would  still  remain  the  impossibility  of  accounting  for 
the  fact  that  such  conversion  invariably  ceased  at  the 
precise  moment  when  each  atom  or  molecule  had  been 
supplied  with  its  due  amount  of  energy. 

In  view  of  all  this  it  is  strange  to  read  in  the  writ- 
ings of  distinguished  physicists  sentences  like  these: 
"The  only  real  things  in  the  physical  universe  are 
matter  and  energy,  and  of  these  matter  is  simply  pas- 
sive,"* and,  "We  see  that,  whereas  (to  our  present 
knowledge  at  least)  matter  is  always  the  same,  though 
it  may  be  masked  in  various  combinations,  energy  is 
constantly  changing  the  form  in  which  it  presents  it- 
self. The  one  is  like  the  eternal,  unchangeable  Fate 
or  Necessitous  of  the  ancients;  the  other  is  Proteus 
himself  in  the  variety  and  rapidity  of  its  transforma- 
tions." f 

There  is  little  doubt  that  the  principle  of  the  con- 

*  The  Unseen  Universe,  §  104.  f  /&.,  §  103. 


CONCLUSION.  <*       307 

servation  of  energy  will  prove  to  be  the  great  theoreti- 
cal solvent  of  chemical  as  well  as  of  physical  phenomena ; 
but  thus  far,  at  least,  the  endeavor  to  express  the  laws 
of  chemical  action  in  terms  of  mass  and  motion  or 
kinetic  energy  has  been  as  abortive  in  chemistry  as  in 
physics.  To  what  extent  it  may  be  possible,  hereafter, 
to  bring  the  phenomena  of  chemical  action  within  the 
dominion  of  the  mechanical  laws  controlling  the  inter- 
action of  solids,  it  is  difficult  to  determine.  There  are, 
however,  several  well-known  facts  which  appear  to  in- 
dicate that,  whatever  be  the  nature  of  chemical  energy, 
it  can  hardly  result  from  the  impact  of  solid  particles. 
The  chemical  energies  of  the  elements  are  proportional 
neither  to  their  masses  as  measured  by  their  weights, 
nor  to  their  volumes ;  and  their  mechanical  equivalents 
are  so  enormous  as  to  seem  out  of  all  analogy  to  ordi- 
nary mechanical  action.  In  1856  W.  "Weber  and  E. 
Kohlrausch  published  the  results  of  a  series  of  investi- 
gations by  which  they  had  sought  to  arrive  at  a  mechan 
ical  measure  for  the  intensity  of  a  galvanic  current. 
They  applied  these  results  to  the  electrolytic  decom- 
position of  water,  so  as  to  determine  the  energy  repre- 
sented in  the  chemical  union  of  hydrogen  and  oxygen. 
And  they  announced  their  conclusion  in  the  following 
words  *  :  "If  all  the  particles  of  hydrogen  in  one  milli- 
gramme of  water  contained  in  a  column  of  the  length 
of  one  millimetre  were  attached  to  a  string,  the  parti- 
cles of  oxygen  being  attached  to  another  string,  each 
string  would  have  to  be  under  a  tension,  in  a  direction 
opposite  to  that  of  the  other,  of  2,956  cwt.  (1 47,830 
kilogrammes),  in  order  to  effect  a  decomposition  of  the 
water  with  a  velocity  of  one  milligramme  per  second." 
And,  looking  to  the  equivalents  of  chemical  energy  in 

*  Pogg.  Ann.,  vol.  xcix,  p.  24. 


308  CONCEPTS  OF  MODERN  PHYSICS. 

terms  of  units  of  heat,  it  has  been  found  that  the  com- 
bination of  a  gramme  of  hydrogen  with  35*5  grammes 
of  chlorine,  so  as  to  form  36'5  grammes  of  hydrogen 
chloride,  is  attended  with  the  liberation  of  an  amount 
of  heat  by  which  the  temperature  of  24  kilogrammes  of 
water  would  be  raised  one  degree ;  inasmuch,  therefore, 
as  the  heat  required  to  raise  one  kilogramme  of  water 
one  degree  is  mechanically  equivalent  to  425  kilogram- 
metres,  the  formation  of  36*5  grammes  of  hydrogen 
chloride  gives  rise  to  a  power  by  which  a  weight  of 
10,000  kilogrammes  can  be  raised  to  the  height  of  one 
metre  in  a  second. 


INDEX. 


ABERRATION,  96. 

Absolute,  the,  159. 

Actio  in  distans,  63,  145. 

Adams,  lunar  acceleration,  61. 

vEther,  113. 

Akin,  history  of  force,  76. 

Allotropy,  302. 

Ampere,  atoms,  84. 

Anderssohn,  mechanics  of  gravita- 
tion, 58. 

Antipodes,  144. 

Arago,  gravitation,  60,  63. 

Aristotle,  dynamical  theory  of  mat- 
ter, 160. 
hypotheses,  121. 
maxim  of,  158. 

Astronomy,  96. 

Atomic  theory,  28,  84. 

Atomic  volume,  174. 

Atoms,  elasticity  of,  41,  42. 

Avenarius,  philosophy  as  thought 
in  conformity  to  principle  of 
least  action,  291. 

Avogadro,  law  of,  34,  35. 

Axioms,  geometrical,  242. 

Babinet,  cosmogony  of  Laplace,  280. 
Bacon,  matter  containing  germs  of 
all  natural  virtue,  essence,  and 
action,  154. 
mathematics,  158. 
Bain,  explanation,  105. 
hypotheses,  116. 
matter,  force,  and  inertia,  161. 
Baumhauer,  atomic  weights,  173. 
Beltrami,    pseudo-spherical    space, 

213. 
Bernoulli,  Daniel,  motion  as  result 

of  oscillation,  60. 
tides.  61. 

14 


Bernoulli,  John,  gravitation,  55. 
conservation  of  vis  viva,  76. 

Berthelot,  chemical  mechanics,  300 

Berthold,   conservation  of  energy, 
69. 

Berthollet,  law  of  chemical  reac- 
tions, 302. 

Bohn,  conservation  of  energy,  76. 

Boltzmann,  second  law  of  thermo- 
dynamics, 27. 
molecular  complexity,  37. 
kinetic  theory  of  gases,  123. 

Boole,  laws  of  thought,  259. 

Boscovich,  atoms,  84. 

Boyle's  law,  90,  117,  123,  124. 

Bradley,  aberration  of  light,  231. 

Briot,  theory  of  light,  97. 

Brodie,  atomic  theory,  103. 

Buckle,  foundations   of  geometry, 
221. 

Budde,    nebular    hypothesis,    280, 
289. 

Calorific  equivalence  of    chemical 

transformations,  303. 
Cardan,  negative  roots  of  an  equa- 
tion, 268. 
Cauchy,    impenetrability,    91,    92, 

179. 

dispersion  of  light,  93. 
Challis,  gravitation,  56,  60. 
Charles,  law  of,  117,  123,  124. 
Clarke,  F.  W.,  planetary  evolution, 

33. 

Clausius,  molecular  complexity,  37. 
kinetic  theory  of  gases,  37,  299. 
molecular  rotation,  123. 
entropy,  270. 

Clifford,  theory  of  light,  155. 
pangeometry,  213. 


310 


INDEX. 


Coleridge,  bedridden  truths,  82. 
Conceivability,  conditions  of,  138. 
Concepts,  nature  of,  132. 
Conservative  system,  77,  78. 
Cooke,  Avogadro's  law,  34. 

atomic  weights,  173. 
Coriolis,  dispersion  of  light,  93. 
Corpuscular  theory  of  matter,  160. 
Cosmogenetic  speculations,  270. 
Cotes,  matter  without  inertia,  44. 
Cournot,  atoms,  102. 

hypotheses,  113. 

solidity  of  matter,  178. 

mathematical  analysis,  108. 
Croll,  force,  168. 
Curvature,  measure  of,  238. 

D'Alembert,  gravitation,  56,  60. 
mathematics,  249. 
geometry,  243. 
Delbceuf,    filiation    of    ideas    and 

things,  178. 
Bellingshausen,  159. 
De   Morgan,   algebraic    quantities, 

267. 
Descartes,   as   a   philosopher    and 

man  of  science,  69. 
conservation  of  motion,  70. 
homogeneity  of  matter,  30. 
matter    nothing    but    extension, 

171,  228. 

mechanical  theory,  16. 
relativity  of  motion,  187. 
Diogenes  of  Apollonia,  69. 
Du  Bois-Reymond,  mechanical  the- 
ory, 21,  23. 
gravitation,  58. 
Duehring,  principles  of  mechanics, 

286. 

Dulong  and  Petit,  law  of,  37,  300. 
Dumas,  atomic  weights,  33,  173. 
Dynamical  theory  of  matter,  160. 

Eddy,  second  law  of  thermodynam- 
ics, 27. 
Elasticity  of  atoms,  40. 

of  vortex  rings,  43. 

resulting  from  rotation  generally, 

45. 

Empedocles,  69. 
Endothermic  combination,  301. 
Energy,  conservation  of,  66,  304. 
Energy,  kinetic,  29,  66,  78,  79. 


Energy,  potential,  29,  66,  67. 

dissipation  of,  271. 
Epicurus,  69. 
Erdmann,    Benno,    dimensions    of 

space,  256. 

Euclid,  geometrical  axioms,  243. 
Euler,  gravitation,  55. 

relativity  of  motion,  192. 
Evolutionism,  158. 
Exothermic  combination,  302. 
Explanation,  nature  of,  105. 

Faraday,  atoms,  84,  160. 

Fatio  de  Duillers,  gravitation,  63. 

Fechner,  atoms,  84. 

Fermat,  16. 

Fichte's  idealism,  235. 

Fictions,  logical,  296. 

Force,  definitions  of,  166,  167. 

measure  of,  73. 

Fresnel,  polarization  of  light,  94. 
Fries,  solidity  of  matter,  182. 
Fritsch,  gravitation,  59. 

Galilei,  laws  of  motion,  79. 
Gases,  diffusion  of,  90. 

kinetic  theory  of,  104. 
Gassendi,  69. 
Gauss,  law  of  motion  under  least 

constraint,  186. 
transcendental  geometry,  208. 
Geometry,  243. 

transcendental,  207. 
Geometrical  axioms,  242. 

inductive  origin  of,  according  to 

Mill,  217. 
George,  159. 
Graham,  Thomas,  equality  of  atoms, 

31. 

hydrogen,  a  metal,  136. 
Grassmann,  dimensions  of  space  not 
deducible  from  laws  of  thought, 
260. 
Guthrie,  gravitation,  59. 

Haeckel,  mechanical  theory,  20. 

Halley,  acceleration  of  the  moon's 
mean  motion,  61. 

Hall,  Asaph,  satellites  of  Mars,  284. 

Halstead,  bibliography  of  hyper- 
space,  208. 

Hamilton,  Sir  William,  cognition, 
105. 


INDEX. 


311 


Hamilton,  Sir  William,  hypotheses, 
116. 

definition  of  concept,  132. 

relation  of  judgments  to  concepts, 
135,  136. 

space,  234. 

principle  of  varying  action,  27. 

conical  refraction,  115. 
Hartmann's  philosophy,  159. 
Heat,  conversion  of,  into  chemical 

energy,  305. 

Hegel's  philosophy,  158. 
Helmholtz,  mechanical  theory,  18. 

vortex  motion,  39,  43. 

electro-magnetic  theory  of  light, 
97. 

all-quality  relation,  184. 

transcendental  geometry,  208. 

conceivability  of  pseudo-spherical 

space,  243. 

Henrici,  geometry,  240. 
Herakleitos,  136. 
Herbart,  relation  of  judgments  to 

concepts,  136. 

Herschel,  Sir  William,  satellites  of 
Uranus,  284. 

Sir  John,  aether,  114. 
Hipparchian  cycle,  109. 
Hobbes,  mechanical  theory,  16. 

sensation,  130. 

Hudson,  wave-theory  of  light,  114. 
Hunt,  T.  Sterry,  solution,  136. 
Huygens,  mechanical  theory,  17. 

gravitation,  55. 

conservation  of  v is  viva,  76. 

Inertia  of  matter,  29,  52, 149,  160. 
Isenkrahe,  gravitation,  49. 

Jevons,  scientific  knowledge,  105. 
nature  of  hypotheses,  111. 
spectroscopy  of  gases,  127. 

Kant,  measure  of  force,  72. 

his  anticipation  of  scientific  dis- 
coveries and  theories,  195. 

pangeometry,  195. 

absolute  center  of  gravity  of  the 
universe,  195. 

nature  of  space,  227,  232. 

nebular  hypothesis,  280. 
Kekule,  atomic  impacts,  102. 
Kepler,  species  immateriaia,  165. 


Ketteler,  dispersion  of  light,  97. 

Kirchhoff,  mechanical  theory,  18. 
lectures  on  mathematical  physics, 

168. 
rest  a  special  case  of  motion,  203. 

Klein,  Felix,  non  Euclidean  geome- 
try, 230,  246. 

Kohlrausch,  R.,  intensity  of  chemi- 
cal energy,  307. 

Krause,  transcendental  geometry, 
237. 

Kroenig,  kinetic  theory  of  gases, 
42,  104. 

Kundt  and  Warburg,  ratio  of  spe- 
cific heat  of  mercurial  vapor  at 
constant  pressure  to  that  at 
constant  volume,  38. 

Land,  transcendental  geometry,  237. 
Laplace,  tidal  retardation,  61. 

nebular  hypothesis,  280. 

invariable  plane,  288. 

lunar  acceleration,  61. 
Lasswitz,    cosmic   gravitation    and 

heat,  275. 

Lea,  H.  Carey,  atomic  weights,  173. 
Le  Conte,  John,  Mars  and  his  satel- 
lites, 285. 
Leibnitz,  mechanical  theory,  16. 

conservation  of  energy,  69,  72, 81. 

letter  to  Clarke,  81. 

simplicity  of  elements,  185. 

relativity  of  motion,  188. 
Le  Sage,  theory  of  gravitation,  63, 

66. 
Lewes,  G.  H.,  conceivability  as  a 

test  of  truth,  110. 
Liebig,  conceivability  as  a  test  of 

truth,  140. 
Light,  undulatory  theory  of,  92,  113. 

dispersion  of,  92,  97. 

polarization  of,  93. 
Lipschitz,    transcendental   mechan- 
ics, 238. 

Lloyd,  conic  refraction,  115. 
Lobatschewsky,  pangeometry,  213. 
Lorentz,  dispersion  of  light,  97. 
Lotze,  concepts,  133. 
Lucretius,  atoms,  84. 

conservation  of  matter,  69. 

Magnus,  dilatation  of  gases,  117. 
Mansel,  theory  of  conception,  132. 


312 


INDEX. 


Mariotte's  law,  90,  177,  123,  124. 
Mass,  relativity  of,  87,  205. 

conservation  of,  26,  27. 
Mathematics,  108,  249. 
Matter,  indestructibility  of,  26,  86. 

solidity  of,  171. 

Maximum  work,  principle  of,  303. 
Maxwell,  James  Clerk,  mechanical 

theory,  18. 

complexity  of  molecular  constitu- 
tion, 37. 

frictionless  aether,  39. 
vortex  rings,  44. 
kinetic  theory  of  gases,  123,  124, 

299. 

molecules,  126. 
definition  of  force,  166,  169. 
entropy,  272. 

Mayer,  Julius  Robert,  theory  of  me- 
teoric agglomeration,  287. 
Mendelejeff,  atomic  weights,  173. 
Metaphysics,  character  of,  137. 
Meyer,  L.,  atomic  weights,  173. 
Mill,  J.  S.,  nature  of  explanation, 

110,  116. 

relativity  of  knowledge,  130. 
nominalism   and    conceptualism, 

133. 
conceivability  as  a  test  of  truth, 

138. 
inductive   origin  of  geometrical 

axioms,  217. 
Mohr,  gravitation,  58. 

mechanical  theory  of  affinity,  299. 
Molecular  work,  principle  of,  303. 
Momentum,  conservation  of,  77. 

angular,  conservation  of,  77,  277. 
Montaigne,  hypotheses,  121. 
Musschenbroek,  mechanical  theory, 
17. 

Naumann,  thermo-chemistry,  300. 
Nebular  hypothesis,  277. 
Neumann,  C.,  dispersion  of  light, 
97. 

body  Alpha,  198. 

Newcomb,  transcendental  space,  21 2. 
Newton,  Sir  Isaac,  inertia,  39,  1 62. 

gravitation,  52. 

letter  to  Bentley,  53. 

queries,  54. 

impenetrability,  89. 

regula  philosophandi^  110. 


Newton,  Sir  Isaac,  relativity  of  mo- 
tion, 189. 

Norton,  W.  A.,  aether,  114. 
Numbers,  complex,  243. 

Occam's  rule,  121. 

Odling,  composition  and  decompo- 
sition, 302. 

Ohm,  Martin,  relation  of  numbers 
to  geometrical  magnitudes,  267. 

Olbers,  cosmical  heat,  274. 

Pangeometry,  207. 
Parmenides,  69. 

Peacock,  principle  of  the  perma- 
nence of  equivalent  forms,  264. 
Peyrard,  edition  of  Euclid,  243. 
Poinsot,  theory  of  rotation,  278. 

resilience  of  rotating  bodies,  46, 
47. 

mathematical  analysis,  108. 
Polymerism,  302. 
Poisson,  inertia,  164. 

polarization  of  light,  94. 
Pott,  development  of  language,  175. 

Quantities,  imaginary,  263. 
Quaternions,  theory  of,   260,   264, 
267. 

Rankine,   second    law  of    thermo- 
dynamics, 27. 
definition  of  force,  170. 
finitude  of  material  universe,  273. 

Rayleigh,  Lord,  electro  -  magnetic 
theory  of  light,  97. 

Realism,  mediaeval,  150. 

Redtenbacher,  dispersion  of  light, 

97. 
absolute  weights  of  atoms,  206. 

Reductio  ad  absurdum,  147. 

Regnault,  expansion  of  gases,  90, 
117. 

Relativity,  principle  of,  134,  183.  ^ 

Riemann,  dissertation  on  hypothe- 
ses which  lie  at  the  base  of  ge- 
ometry, 209,  248. 

Rohmer,  159. 

Rudberg,  expansion  of  gases,  117. 

Sachs,  Uranus,  169. 
Schelling,  matter  the  seed-corn  of 
the  universe,  154. 


INDEX. 


313 


Schelling,  dynamical  theory  of  mat- 
ter, 160. 

Schlegel,  theory  of  space,  260. 
Schleicher,  Darwin  and  language, 

291. 

Schopenhauer'sphilosophy,159,235. 
Schramm,  matter  and  motion,  59. 
Sigwart,  theory  of  conception,  133. 
Somoff,  definition  of  force,  167. 
Space,  nature  of,  235,  237. 
properties  of,  240. 
relativity  of,  204. 
a  concept,  235. 
non-homaloidal,  207. 
Spiller,  aether,  164. 
Spencer,  Herbert,  equality  of  ele- 
ments, 31. 
symbolic   character  of    thought, 

135. 
conceivability  as  a  test  of  truth, 

138. 
Spinoza,  philosophy  of,  70. 

proposition   that  the   order   and 
connection  of  ideas  are  identi- 
cal with  those  of  things,  177. 
Stas,  atomic  weights,  173. 
Stefan,  kinetic  theory  of  gases,  123, 

124. 

Stevinus,  16. 
Stewart,  Balfour,  gravitation,  58. 

expansion  of  gases,  117. 
Stumpf,  origin  of  concept  of  space, 

234. 
Sylvester,  transcendental  space,211. 

Tait,  P.  G.,  gravitation,  58,  66. 
force  and  momentum,  166. 
transcendental  space,  210. 
degradation  of  energy,  272. 
entropy,  272. 
Tauschinsky,  theory  of  conception, 

133. 

Taylor,  kinetic  theory  of  gravita- 
tion, 58,  66. 

Thermometer,  graduation  of,  118. 
Things  per  se,  159. 
Thomson,  J.   J.,  electro -magnetic 
theory  of  light,  97. 


Thomson,  Sir  William,  atoms,  101. 

elasticity  of  molecules,  42. 

vortex  rings,  43. 
Thought,    relation    of,    to    things, 

129.  ^^ 

Time,  relativity  of,  204.  l*^ 
Tyndall,  Belfast  address,  153. 
J  Liverpool  address,  154. 

atoms,  154. 

Univer salia  ante  rem,  150. 

in  re,  150. 
Universe,  finitude  of,  274. 

Varignon,  gravitation,  63. 
Virtual  velocities,  principle  of,  78. 
Vis  viva,  74. 
conservation  of,  75. 

Wallace,  adaptation  by  general  law, 

195. 
Walter,  Arwed,  planetary  molecule, 

122. 
Weber,  W.,  intensity  of  chemical 

energy,  307. 
Werder,  159. 

Whately,  theory  of  conception,  135. 
Whewell,  conceivability  as  a  test  of 

truth,  138. 
inertia,  162. 
force,  166. 

Wittwer,  elasticity  of  atoms,  40. 
Wrede,  aether,  114. 
Wright,   R.   A.,   allotropy  of  ele- 
ments, 33. 

Wundt,  mechanical  theory,  19. 
equality  of  atoms,  31. 
theory  of  conception,  133. 
hypotheses,  105. 
cosmological  problem,  271. 

Young,  inertia,  162. 
force,  166. 

Zoellner,  requisites  of  valid  hypoth- 
esis, 109. 

Newton's  letter  to  Bentley,  54. 
concept  offeree,  167. 


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